Double-trace flows and the swampland

We explore the idea that large N, non-supersymmetric conformal field theories with a parametrically large gap to higher spin single-trace operators may be obtained as infrared fixed points of relevant double-trace deformations of superconformal field theories. After recalling the AdS interpretation and some potential pathologies of such flows, we introduce a concrete example that appears to avoid them: the ABJM theory at finite k, deformed by ∫O2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\displaystyle \int {\mathcal{O}}^2} $$\end{document}, where O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document} is the superconformal primary in the stress-tensor multiplet. We address its relation to recent conjectures based on weak gravity bounds, and discuss the prospects for a wider class of similarly viable flows. Next, we proceed to analyze the spectrum and correlation functions of the putative IR CFT, to leading non-trivial order in 1/N. This includes analytic computations of the change under double-trace flow of connected four-point functions of ABJM superconformal primaries; and of the IR anomalous dimensions of infinite classes of double-trace composite operators. These would be the first analytic results for anomalous dimensions of finite-spin composite operators in any large N CFT3 with an Einstein gravity dual.


Introduction
The paradigmatic example of AdS/CFT [1][2][3] is a duality between two theories: a theory of AdS quantum gravity whose low-energy limit includes a weakly coupled Einstein gravity subsector; and a CFT with many degrees of freedom ("large N "), and a sparse spectrum of local, single-trace operators of spin no greater than two.The absence of parametrically light single-trace operators of higher spin is often phrased as a holographic gap condition, ∆ gap 1, where ∆ gap is the dimension of the lightest spin-four single-trace operator [4].As a matter of consistency, these conformal field theories must, among other properties, be unitary, crossing-symmetric and causal; must they also be supersymmetric?
There are recent conjectures in the affirmative, motivated in part by an absence of explicit constructions.It was argued in [5,6] that weak gravity bounds on the tension of charged objects in quantum gravity may only be saturated when a BPS condition is obeyed; combined with known instabilities of non-BPS branes in AdS spacetimes, [5,6] argued that, indeed, sparse non-supersymmetric CFTs with large N and large gap do not exist.This connection arises because the weak gravity bounds are meant to hold in theories whose low-energy gravitational sector is described by general relativity, a feature which is automatically implied by the CFT spectral condition ∆ gap 1 [7][8][9].In this paper, we explore a general approach to the construction of large N , large gap, non-supersymmetric CFTs, and investigate specific examples that avoid some well-known pathologies that are both perturbative and non-perturbative in N .The main idea is to impose supersymmetry-breaking boundary conditions in AdS.On the CFT side, we perturb a UV superconformal field theory by a relevant, supersymmetry-breaking, double-trace deformation, which flows to an IR CFT that still obeys the gap condition.The construction is on firmest footing when the UV SCFT has no symmetry-preserving exactly marginal operators.While not all such flows may lead to a viable IR fixed point, we present a large class of examples in three spacetime dimensions that appear to be especially promising. 1This construction may be viewed as a potential counterexample to the CFT conjecture of [5,6], but is consistent with their sharpened version of the weak gravity bounds as applied to flat space quantum gravity.
In Section 2, we introduce the basic idea of a supersymmetry-breaking double-trace flow from a large N , large gap SCFT; make a connection to the conjectures of [5,6]; list some pathologies that a consistent construction must avoid; and discuss the most basic constraints on such flows from superconformal representation theory in various spacetime dimensions.
In Section 3, we introduce a specific class of examples: namely, the ABJM theories [18] at finite k deformed by O 2 , where O is the superconformal primary in the stress-tensor multiplet.We argue for their viability at large but finite N , and address a potential issue on the moduli space of vacua.We also discuss a preliminary proposal for using double-trace deformations of more general three-dimensional CFTs to generate other non-supersymmetric theories with a large gap.
In Section 4, having made a concrete proposal for a non-supersymmetric, large gap CFT, we perform some quantitative computations of its observables.We show that certain classes of correlation functions vanish in the IR at leading order in 1/N , and very briefly discuss the implications for thermal physics.We then study the change under RG flow of an infinite class of four-point functions of ABJM superconformal primaries O p .This calculation generalizes the approach of [19].By performing the conformal block decomposition of those results, we extract, in Section 4.3, the leading-order change in the anomalous dimensions of spinning double-trace operators O p ∂ µ 1 . . .∂ µ O p .Moreover, using the non-renormalization properties of UV-protected supermultiplets, we can extract the anomalous dimension of several classes of double-trace operators at the IR fixed point, not only their change under RG flow.The main results may be found in (4.34), (4.38)- (4.40), and (4.43).These would be, to our knowledge, the first analytic computations of anomalous dimensions of finitespin double-trace operators in any large N , three-dimensional CFT with an Einstein gravity dual, supersymmetric or otherwise.Our technique for analytically deriving these anomalous dimensions applies to any IR fixed point obtained by double-trace flow from a CFT with a greater number of supersymmetries, including, in particular, IR SCFTs that preserve a fraction of the UV supersymmetry.
In Section 5, we conclude with some comments on future work.Appendices A-B include some technical details needed for Section 4, while Appendix C gives a brief historical recollection of the attempt to construct non-supersymmetric, large gap CFTs by orbifolding N = 4 super-Yang-Mills.
Before proceeding, it is worth being more precise about our definitions, and our interpretation of the conjectures of [5,6] that motivated this work.In general, a CFT with a weakly coupled gravity dual belongs to a sequence of CFTs with central charge C T parameterized by N .The sequence exists above some critical value of N and admits a N → ∞ limit with a finite spectral density and operator product expansion.The prototypical holographic CFT, whose gravity sector is general relativity, obeys the further constraint ∆ gap 1, and the "sparseness" condition of polynomially-bounded growth of low-spin operators.(This can include towers of KK modes from transverse manifolds with AdS-scale curvatures.)It is this class of CFTs whose existence we are addressing.It should be emphasized that the question of whether holographic CFTs can have a parametrically large gap ceases to make sense at small enough N : these theories always obey a parametric hierarchy ∆ gap √ C T .(This was recently proven in CFT in [20].)In string theory terms, M string M P lanck .For finite N and ∆ gap , UV-finiteness demands corrections to the gravitational action beyond general relativity, at which point it is unclear what becomes of the conjecture of [5,6] then.

Basic Idea
Let us first quickly recall the definition of a double-trace flow.Consider a large N CFT d which contains a scalar conformal primary O of conformal dimension ∆ < d/2.If we deform the action by In this paper, we take the UV theory to be a superconformal field theory with large N and a large gap, and O to be a superconformal primary.
operator is identical in the UV and IR.At the non-planar level, conformal dimensions and OPE coefficients are modified, both for single-trace and multi-trace operators.The change in the double-trace spectrum was recently analyzed in [19], and we will reprise those results later in this paper.
In the holographic context [21][22][23][24][25][26][27][28], in which O is dual to a scalar field φ of mass squared m2 = ∆(∆ − d) in AdS units, there are two choices of normalizable boundary conditions when −d 2 /4 ≤ m 2 ≤ −d 2 /4 + 1: Each of these corresponds to a unitary conformal dimension at one end of the RG flow triggered by (2.1).The flow is not visible in the bulk as a soliton interpolating between two AdS vacua with a macroscopic difference in curvatures, due to 1/N suppression.We now consider this deformation in a top-down setting.Consider a SUSY AdS d+1 × M compactification of string or M-theory.In the limit in which the bulk theory is a weakly coupled supergravity and M is of AdS size, as in Freund-Rubin compactifications, the dual SCFT has a parametrically large gap to single-trace operators of spin greater than two, and the light spectrum is sparse. 2If the Kaluza-Klein spectrum on M contains scalar fields with masses in the aforementioned range, SUSY may require alternate quantization.This happens, for instance, for bottom components of chiral multiplets of unit R-charge in 4d N = 1 SCFTs (where ∆ = 3/2), or of flavor current multiplets in 3d N = 2 SCFTs (where ∆ = 1).In such cases, we may deform the dual SCFT as in (2.1), where O is the conformal primary operator dual to such a scalar field.Such a double-trace deformation will generically break all SUSY in the IR.Thus, by choosing the SUSY-breaking ∆ + boundary condition in AdS, we are studying a non-SUSY, large N fixed point with ∆ gap 1.Let us make contact with [5,6].Their proposed sharpening of the weak gravity conjecture states that in a consistent theory of quantum gravity, the weak gravity bounds on chargeto-mass ratios may only be saturated by BPS objects.AdS spacetimes suffer from a brane nucleation instability for non-BPS branes with a sufficiently large charge-to-tension ratio [30,31], which destroys the near-horizon region of a stack of branes in asymptotically (locally) flat space; [5,6] take this as evidence that non-SUSY CFTs with large N and a large gap do not exist.But the special class of non-SUSY CFTs studied herein -obtained by SUSYbreaking boundary conditions in AdS geometries built from BPS branes -may nevertheless be consistent with the refined bound on charged objects in flat space.In the brane picture -that is, before taking the near-horizon limit -there is no boundary, and hence no notion of SUSY-breaking boundary condition. 3We return to this point in Section 3.

What can go wrong?
Obviously, the question is how robust this construction is away from infinite N .But some pathologies are absent by design.At infinite N the CFT has a unitary spectrum of operator dimensions; in the bulk, the classical theory is perturbatively stable.Moreover, nonperturbative instabilities arising from geometry or topology of M are also absent.
There are, still, various potential pitfalls that may render the IR CFT ill-defined, including i) complex operator dimensions in the 1/N expansion; ii) an unstable vacuum; iii) the breaking of conformal symmetry via nonzero beta functions for marginal or nearly-marginal gauge-invariant operators, which can be generated for single-trace or multi-trace operators.
If the CFT has a conformal manifold, the offending pathology -and its bulk dual -may change as a function of marginal couplings (as in e.g.[33,34]).
On the bulk side, one may likewise find i) bulk tachyons that violate the Breitenlohner-Freedman bound [35].This can occur at tree-or loop-level, but the most dangerous situation is when a tree-level scalar mass sits at the bound, m 2 BF = −d 2 /4: in the absence of SUSY, quantum corrections may push m 2 < m 2 BF ; ii) the development of a runaway direction in the effective potential for probe branes in AdS; iii) non-perturbative instabilities due to the geometry and/or topology of M; iv) non-perturbative instabilities in AdS, such as the potential for brane nucleation of [30,31].

In which spacetime dimensions?
With these issues in mind, let us briefly comment on the viability of this construction in various space-time dimensions, before moving on to the case of d = 3, our main interest.
In d = 4 SCFTs, a ubiquitous class of operators with ∆ < 2 are N = 1 chiral primaries with unit u(1) R -charge, which have ∆ = 3/2.However, finding viable examples of SUSYbreaking double-trace flows to stable IR fixed points in d = 4 seems challenging.The bottom component of a four-dimensional N = 1 current multiplet is a ∆ = 2 superconformal primary, O.The dual scalar field in AdS sits exactly at the BF bound.One can also form doubletrace operators O 2 that are classically marginal global symmetry singlets; such operators are typically marginally relevant, and may develop nonzero beta functions along a conformal manifold.The latter issue also applies to composites made from fermions in N = 1 chiral and anti-chiral multiplets, which have ∆ = 2.An explicit example of a large N , large gap CFT where these problems would arise is the Klebanov-Witten theory, dual to type IIB string theory on AdS 5 × T 1,1 [37].The theory contains an operator of unit R-charge, O 1 , that lives in the (2, 2) of an su(2) A × su(2) B global symmetry.This is the only operator of ∆ < 2. Thus, if we turn on the deformation by choosing the ∆ + boundary condition on the dual bulk scalar field, the CFT will naively flow to a non-SUSY fixed point.This was briefly considered in [25,38,39].But for the reasons given above, the putative IR fixed point is unlikely to exist.(The Klebanov-Witten theory also has an exactly marginal coupling, which can acquire a nonzero beta function in the 1/N expansion after double-trace flow.)On the other hand, the representation theory of osp(N |4), the d = 3 superconformal algebra, is much more favorable, and we focus on this case below.

A SUSY-Breaking Double-Trace Flow from ABJM
We now turn to our main proposal: that a non-SUSY double-trace flow from the ABJM theory leads to a stable IR fixed point.
Let us quickly review the salient aspects of the ABJM theory that we will need.(For more detailed reviews, see e.g.[40][41][42].)The ABJM theory [18] is a parity-invariant, U (N ) k × U (N ) −k Chern-Simons theory coupled to bifundamental matter.For k = 1, 2, the theory has N = 8 SUSY, with so(8) R-symmetry, whereas for k > 2 it has N = 6 SUSY, with su(4) R × u(1) ⊂ so(8) R-symmetry.The nature of the large N limit depends on whether k is held finite.At finite k, the holographic dual is 11d supergravity on AdS 4 × S 7 /Z k , supported by four-form flux.At large k, one may take a 't Hooft limit, k → ∞, N → ∞, with λ = N/k fixed, where, up to numerical prefactors, α √ λ ∼ 1. Upon increasing k while keeping λ 1, the bulk dual remains 11d supergravity up to k5 ∼ N , where it crosses over to type IIA supergravity on AdS 4 × CP 3 .Eventually, for large enough k, stringy effects become parametrically important.It is useful to view the S 7 /Z k as a circle fibered over CP 3 with length 2πL AdS /k; this essentially corresponds to the u(1) symmetry of the CFT.The "central charge" C T of the ABJM theory is where C T is defined by the stress tensor two-point function as with a fixed tensor structure I µνρσ (x) defined in [43].
For any value of k, the bottom component of the stress tensor multiplet is a ∆ = 1 scalar, which we call O.For k = 1, 2, O resides in the 35 c of the so(8) R-symmetry, 5 with Dynkin labels [0020]; we may represent it as O IJ , the symmetric, traceless rank-two tensor of so (8), where I, J = 1 . . .8. In terms of the fundamental scalars X I in the ABJM Lagrangian, For k > 2, O resides in the 15 of the su(4) R-symmetry, with su(4) Dynkin labels [101]; we may represent it as O a b .In either case, from these components, one may form many double-trace operators, preserving varying degrees of R-symmetry.A natural choice is to preserve the full R-symmetry, but break all SUSY.Thus, we propose to consider the RG flow away from the ABJM theory triggered by the R-symmetry singlet O 2 , with all R-symmetry indices contracted: where Let us lay out some arguments for this.First, as mentioned earlier, the AdS 4 × S 7 /Z k geometry admits no tunneling solutions that can be ascribed to the transverse manifold, because it descends from a SUSY compactification.Second, at finite k the ABJM theories form a discrete set, with no exactly marginal gauge coupling.This is one virtue of AdS 4 ×M 7 compactifications in general as compared to AdS 5 × M 5 compactifications.
Another appealing property of the finite k ABJM theories subject to the R-symmetrypreserving double-trace deformation (3.4) is their especially sparse spectrum of light operators: all symmetry-preserving gauge-invariant operators in the IR are irrelevant.More precisely, the IR fixed point has the following two properties, to all orders in 1/N : A There are no global singlet, parity-preserving relevant operators.
B There are no global singlet, parity-preserving marginal operators for k = 1, 3.
B implies that there are no ∆ = 3/2 scalar operators, which would be dual to BF boundsaturating scalars.(If there were, we could use them to form a classically marginal singlet double-trace operator.)Together these imply the complete attractiveness of the double-trace RG flow and the absence of conformal symmetry breaking.
To show A and B, we examine the spectrum of gauge-invariant operators of the ABJM theory.The spectrum at large N may be obtained by starting from the KK reduction on S 7 [45,46], and restricting to Z k -invariant modes.See [47] for a summary.These are the modes with u(1) charge 0 mod k. 6 The scalar spectrum on S 7 is comprised of a tower of KK modes in so(8) representations [00p0], where p = 2, 3, . ... The dual operators are singletrace, superconformal primaries which we call O p , with ∆ p = p/2.Each of these is a bottom component of a 1/2-BPS superconformal multiplet, namely, the B 1 [0] (0,0,p,0) p/2 multiplets in the notation of [36].The p = 2 multiplet is the stress tensor multiplet, which contains O 2 ≡ O; we list its content in Table 1.Under the branching so(8 We now look to construct relevant and marginal scalar operators at the IR fixed point that are singlets under all global symmetries.We have broken SUSY completely, but kept the full R-symmetry.The theory is also parity-invariant.So we are looking for parity-even R-singlets.As described above, all single-trace scalars are charged under the R-symmetry, Table 1: The conformal primaries of the stress-tensor multiplet in a 3d, N = 8 SCFT, together with their so(3, 2) × so( 8) quantum numbers and their positions in the multiplet.This is the multiplet in the notation of of [36].The superconformal primary is O.In a theory with N = 6 SUSY, the first three columns are identical.so only multi-trace operators can be R-singlets.In the UV, before the RG flow, the only operators that can possibly be used to form marginal or relevant multi-trace scalar singlets have ∆ = 1, 3/2, 2. Taking k = 1, 3, which removes the potentially problematic O 3 from the spectrum, the list of such operators is short: • O, with ∆ = 1 in the 35 c , and its superconformal descendants with ∆ ≤ 2. These include spin-1/2 fermions ψ, with ∆ = 3/2 in the 56 v , and scalars, with ∆ = 2 in the 35 s -see Table 1.
Due to so(8) selection rules, the only relevant singlet comprised of these constituents is O 2 , our deforming operator.In the UV, there is also a nearly marginal triple-trace operator obtained from O 3 (projecting onto the singlet part).However, the RG flow (3.4) takes ∆ → 2 + O(N −3/2 ), for all components of O.This immediately implies that for generic k, there are no relevant singlets in the IR. 7Likewise, the only candidate (nearly-)marginal singlets in the IR are the two-fermion operators, :ψψ:, projected onto the singlet.However, this operator is parity odd. 8Thus, we have shown Properties A and B. These imply that the RG flow (3.4) leads to a stable fixed point, and the IR CFT admits no symmetry-preserving relevant flows.
Having established the absence of relevant and marginal global singlet operators, let us however note that such operators would not have been expected to spoil the existence of the IR fixed point anyway.Given some set of operators O i with ∆ i − d = i 1 and corresponding couplings g i , ordinary conformal perturbation theory admits both trivial (g i = 0) and non-trivial (g i ∼ i ) fixed points.For either sign of i , the trivial fixed point is guaranteed to exist.This should be contrasted with the case of non-SUSY orbifolds of N = 4 SYM in the 't Hooft limit, in which, due to the exactly marginal 't Hooft coupling, nearly-marginal global singlets develop nonzero beta functions to any order in perturbation theory [49][50][51].Note, though, that even when there are flat directions in a theory with nearly-marginal singlets, the addition of a relevant deformation, like O 2 , may still lead to a stable IR fixed point.This happens in e.g.[37,52,53]; while these examples retain some SUSY in the IR, perhaps the same also happens in the 't Hooft regime of the ABJM theories after our SUSY-breaking double-trace deformation.

A general prescription for 3d CFT
There are obvious variations of the above, including extension to the [54], or the introduction of double-trace deformations that preserve only a subgroup of so (8).More interesting are generalizations beyond ABJ(M).Consider an AdS 4 × M 7 solution of 11d supergravity, where M 7 is a Sasaki-Einstein manifold.The dual N = 2 SCFT is, like ABJM, isolated.A preliminary proposal for generating stable, IR non-SUSY CFTs is to change the AdS boundary conditions for all single-trace scalar operators with ∆ < 3/2.That is, for every such operator O i with ∆ i < 3/2, deform the UV SCFT action by for some couplings g i .Following our ABJM discussion, the question of IR stability boils down (modulo issues we have not exorcised yet) to the question of whether there are any single-trace scalar operators with ∆ = 3/2.Examples abound.In d = 3, the bottom component of N = 2 flavor multiplets is a ∆ = 1 scalar.(N = 1 flavor multiplets do not contain scalars, so N = 1 SCFTs may, but need not, contain a ∆ < 3/2 scalar with which to flow.)A well-studied N = 2 example is the 11d supergravity compactification on AdS 4 × M 111 [55,56], dual to a quiver gauge theory with three nodes [57].This manifold has isometry group G = su(3) × su(2) × u(1) R × u(1) B , where the u(1) B is a baryonic or "Betti" vector multiplet.From the analysis of the KK spectrum in [55] (see also Table 5.2 of [13]), one can check that the CFT contains no ∆ = 3/2 singletrace scalar operators. 9It would be worthwhile to examine this proposal further.

Comments Connection to other proposals
The double-trace technique considered in this paper is "milder" than other SUSY-breaking constructions that break SUSY in the bulk, not just by boundary conditions.In Appendix C, we briefly recall the story of one of the most well-studied -and ultimately unsatisfactory -non-SUSY constructions, namely, the type IIB orbifolds of the form AdS 5 × S 5 /Γ, dual to non-SUSY orbifolds of 4d N = 4 SYM.
In [17], a morally similar construction (inspired in part by [58,59]) was suggested for the k = 1 ABJM theory, in which the so(8) R-symmetry is gauged and augmented by a Chern-Simons term.This may be implemented by a double-trace deformation δS CFT = d 3 xJ µ J µ , where J µ is the R-symmetry current, which is induced holographically through a mixed boundary condition on the bulk gauge field [60].Though the IR fixed point may indeed exist, it has more potentially problematic operators whose dynamics may destabilize the theory and/or drive it to non-unitarity (such as the triple-trace singlet O 3 discussed in [17]); it also would have the relevant singlets O 2 , thus making the RG flow less stable.Our proposal is simpler, and eliminates these operators, as described above.

Moduli space of vacua
A potential issue with this class of theories is the stability of the moduli space of vacua.In the ABJM theory on R 3 , the effective potential on the moduli space vanishes.In the IR, in the absence of SUSY, these flat directions will presumably be lifted at finite N ; in principle, the origin of moduli space could cease to be a minimum, or runaway instabilities could develop in the 1/N expansion. 10Definitively understanding the fate of the moduli space appears to be highly involved, but let us make some observations. 11 Intuitively, the deformation (3.4) appears to lift the moduli space.With g > 0, it is a stabilizing quartic potential for all ABJM scalars, that grows as one flows toward the IR.It is nevertheless possible that, in the deep IR, the minimum is pushed away from the origin, or worse, by higher order effects in 1/N .(If there is indeed a minimum away from the origin, this would be an interesting non-SUSY, non-conformal field theory to study.) At finite k, the CFTs are inherently strongly coupled, so one must resort to a bulk Mtheory computation.The essential question of whether a nucleation instability [30,31] occurs boils down to whether the effective potential for probe M2-branes in AdS 4 × S 7 /Z k , with the modified boundary condition for φ (dual to O), is attractive or repulsive.(A related Accordingly, while the IR fixed point may be stable, it is thus not a "dead-end" CFT. 10 See [33,61,62] for discussions of similar instabilities in the non-SUSY N = 4 SYM orbifold context.In [51], it was shown that in fixed lines of 4d CFTs with adjoint matter, Coleman-Weinberg instabilities exist if and only if conformal symmetry is broken via nonzero beta functions.If this equivalence extends to the present case (though we know no reason this would be so), our previous arguments about the spectrum imply an absence of moduli space instabilities for the ABJM double-trace deformation. 11We thank O. Aharony for raising this issue and for valuable discussions about it.
approach would be to compute the force between two probe M2-branes using the ∆ = 2 boundary condition for φ.)The only contributions to the potential that differ from the SUSY case must involve φ propagators.At leading order in 1/N , one can compute the "self-energy" correction due to the emission and re-absorption of φ from the brane, given the ∆ = 2 boundary condition.This should yield the leading order effect of the field theory potential (3.4) in the IR.(A related calculation was performed in [63].)We expect that this leaves flat the O = 0 subspace of the moduli space, so to determine whether this is lifted requires a higher-loop bulk computation.Unfortunately, this is no longer an AdS 4 supergravity computation, as it involves all of the scalar KK modes φ p , dual to O p .For instance, the emission/re-absorption of φ p from the brane receives a loop correction, because all φ p couple to φ through loops.A vev for O p would also seem to receive linear contributions from one-loop tadpole diagrams, where a φ loop attaches to a φ p propagator attached to the probe; however, we note the encouraging feature that all cubic couplings φ p φ 2 vanish. 12here are likely other effects to consider as well; we leave a systematic exploration for future work.
4 Spectrum and Operator Products of the IR CFT The IR CFTs described here have a rigid structure despite the lack of SUSY.The spectrum of local operators is integer-or half-integer spaced (depending on the parity of k), to leading order in 1/N .Their operator products obey strict selection rules imposed by so(8) global symmetry.The three-sphere free energy, F = − log Z S 3 , can also be written to several subleading orders in 1/N [64]: F ABJM is known exactly from the ABJM matrix model [65].In particular, it includes terms of order N ±1/2 and log N that are identical in the IR CFT: higher loop effects start at O(N −3/2 ).It is somewhat remarkable that these subleading terms representing quantum effects in M-theory are robust to SUSY-breaking boundary conditions.Below we derive some new results on the CFT data at the putative IR fixed points.In Section 4.1, we show that even the leading large-N contribution to certain single-trace correlators flows to zero in the IR, to leading order in 1/N .In particular, this is true for "extremal" n-point correlators, and for the non-extremal three-point function O 2 O 2 O 2 . 13n Section 4.2, we compute the change in the connected four-point functions O p O p O p O p , with p = 2, between UV and IR.From this we extract, in Section 4.3, the leading-order change in the conformal dimensions of double-trace operators O p ∂ 2n ∂ µ 1 . . .∂ µ O p (modulo an issue of operator mixing, which we explain).However, the real power of this calculation comes in considering the consequences of so(8) global symmetry.There exist several families of double-trace operators, one for each so(8) representation appearing in the product O p ⊗O p .In the UV, some of these operators reside in SUSY-protected multiplets, and thus have vanishing anomalous dimensions to all orders in 1/N .But in the IR, absent SUSY, these multiplets are no longer protected.Therefore, for these so(8) representations, the change in the anomalous dimension between UV and IR equals the IR anomalous dimension!Thus, our computation allows us to read off some analytical double-trace spectral data about the IR CFT.

Extremal correlators vanish after double-trace flow
An extremal correlator is defined by the condition These were studied mainly in the N = 4 SYM context in [66], and then in the ABJM context in [67].We now demonstrate two simple vanishing conditions of extremal correlators under double-trace flow, and of the non-extremal correlator O 2 O 2 O 2 .

Three-point functions
First, for completely general double-trace flows, an extremal three-point function that involves O, the operator that triggers the flow, becomes zero in the IR to leading order in 1/N : General double-trace flows: For the flow from ABJM, this implies that an infinite set of three-point functions vanishes at leading order in the IR: for all superconformal primaries O p for any integer p in the ABJM spectrum, where the UV result may be read off from [68] (we have used a unit normalization of the two-point functions O p O p ).
It is straightforward to prove (4.3).For simplicity, consider the three-point extremal correlators ΦOO , where ∆ Φ = 2∆ O in the UV.In the IR, we trade O for its Hubbard-Stratanovich field σ [39] inside correlation functions.In Appendix A, we show that the IR OPE coefficient, C IR Φσσ , is where C OO is the norm of O.The denominator of the last factor implies that the UVextremal correlator vanishes in the IR: C IR Φσσ = 0.The analogous calculation was done for ΦΨO where ∆ Φ = ∆ Ψ + ∆ O in [19], which, being extremal, can also be seen to vanish in the IR.
This can be understood holographically using well-known facts about extremal correlators [66].The bulk fields participating in extremal CFT three-point functions have a vanishing bulk cubic coupling, regardless of the boundary condition.In the UV, multiplying this zero by the infinity from the AdS integral gives a finite result (a more formal treatment involves subtle boundary terms).But in the IR -that is, after changing quantization of the bulk field dual to O -the AdS integral does not produce an infinity because the correlator is no longer extremal.Thus, one gets zero in the IR.This can also be understood yet another way, by thinking about operator mixing in the identification of bulk fields with CFT operators.A nonzero CFT extremal correlator ΦOO is only consistent with a vanishing bulk coupling if the bulk field Φ bulk is dual not only to Φ, but to a linear combination for some c such that the three-point function of bulk modes, Φ bulk O bulk O bulk , vanishes.In the IR, ∆ O → d − ∆ O , and this operator mixing is not allowed: thus, the holographic identification is Φ bulk := Φ, and the CFT correlator vanishes.

Self-coupling of O 2
Let us also point out that the three-point function of O 2 vanishes in the IR: again assuming unit normalization of O 2 , This correlator is not extremal, but shares the feature that the bulk cubic vertex for φ 2 vanishes; the nonzero result in the UV is due to a compensating factor Γ ∆ 1 +∆ 2 +∆ 3 −d 2 in the AdS three-point scalar integrals [69].(See [70] for a proper treatment of boundary terms in N = 8 supergravity that yields the correct result.)In the IR where ∆ → 2 + . .., this gamma function becomes finite, so the CFT three-point function vanishes.The analogous statement is true for the three-point function of φ 2 in the large N critical O(N ) model [71][72][73].

Application: Thermal one-point functions
The The leading term in the IR comes from the cubic self-coupling of O 4 because neither of the other IR operators with ∆ ≤ 2 -in particular, the ∆ = 3/2 fermion and ∆ = 2 scalar in the stress tensor multiplet -produce a 294 c in their so(8) tensor product [74].Moreover, in the IR, all multi-trace operators made of O 2 do not contribute to O 4 S 1 β ×S 2 at leading order in 1/N : for these operators, the leading order contribution comes from (generalized) free field Wick contractions, which always leaves a three-point factor O 2 O 4 O 2 IR = 0. Similar statements are true for the thermal one-point function of O 2 .For instance, at small β, where ψ is the spin-1/2 fermionic operator in the 56 v of so(8) (see Table 1).

n-point functions
Next, in the double-trace flow from ABJM, any n-point extremal correlator involving at least one O 2 also vanishes in the IR to leading non-trivial order in 1/N : where p 1 = 2 + n−1 i=2 p i .Although it is not directly to the question of whether tree-level extremal n-point correlators vanish after double-trace flow, we note for completeness that in ABJM (indeed, in maximally-SUSY CFTs in 3 ≤ d ≤ 6 [75]), extremal correlators of chiral primaries exhibit the factorized form where p 1 = n i=2 p i [67].The mechanism can again be viewed as coming from the admixture of O p 1 with the (n − 1)-trace operator [O p 2 . . .O pn ].Upon flowing to the IR, this vanishes.
The proof of (4.12) adapts the arguments of [66] to this setting.In fact, this was already done in [67].For simplicity, we consider the N = 8 ABJM theories, so we study the four-point function For the double-trace application, we take (say) p 4 = 2. Using the so(8) tensor product (e.g.[75]) and likewise for [00p 3 0] ⊗ [00p 4 0], one sees that the only so(8) representation in both tensor products is [00(p 3 + p 4 )0].The logic of [66] applies verbatim, and all bulk diagrams contributing to these correlators involve at least one vanishing bulk vertex.As explained earlier, it follows that the IR correlator vanishes.

Four-point functions
In [19], the change in connected four-point functions ΦΦΦΦ was computed under general double-trace flows O 2 , for single-trace operators Φ that couple to O.This was used to extract the change in the spectrum of double-trace operators Φ∂ 2n ∂ µ 1 . . .∂ µ Φ in the IR, as well as their OPE coefficients.We can generalize this result to the present case, in which we take Φ to be O p , the superconformal primaries of ABJM.

Setup
The spectrum of superconformal primary operators of the ABJM theory, for any k, includes the infinite tower of 1/2-BPS chiral primaries O p , where p = 2, 4, . .., living in the [00p0] representations of so (8) or its branching into su(4) × u(1).These operators have conformal dimension ∆ = p/2.For concreteness, in the remainder of this section we specialize to k = 1, 2, and hence an so(8) global symmetry, though the results are easily generalized.We may form so(8) invariants by contracting their indices with the null vectors Y I , .Due to SUSY Ward identities relating O 2 to T µν , O 2 couples universally to all operators, so this difference is guaranteed to be nonzero for all p.The rest of the calculation is an extension of so( 8) group theory to the results of [19].
The functional form of two-and three-point functions of O p are determined by so(3, 2) × so(8) symmetry.Let us introduce the following UV correlators, for some constants C pp , C pp2 , where x 12 ≡ |x 1 − x 2 |.We may form the normalizationindependent ratio, This may be computed in various ways (see e.g.[68]) to be Symmetry allows us to write the four-point function at either fixed point in the form where we introduced the internal cross-ratios F p (u, v; σ, τ ) has an expansion in the so(8) representations appearing in the tensor product R p ⊗ R p , where we sometimes employ the shorthand The list of such representations is Representations in the symmetric product have a + b = even, whereas those in the antisymmetric product have a + b = odd.The contribution of each representation to δF p is encoded in a harmonic polynomial of so( 8) which depends on both σ and τ [76,77].These polynomials Y ab (σ, τ ), associated to the representation (ab), obey an orthogonality condition Hence the four-point function enjoys the decomposition Crossing symmetry of F p (u, v; σ, τ ) acts in all four variables.The algorithm for constructing these polynomials, as well as the first several explicit polynomials, can be found in [76,77].
Of particular importance for what follows are the polynomials associated to (00) = [0000], the singlet, and (11) = [0020], in which O 2 lives: In what follows, we will compute δF p (u, v; σ, τ ), the difference between IR and UV connected correlators:

Change in four-point function
As explained in [19], δF p is given by a sum of three terms, each of which computes the change in the contribution of O 2 in a given channel.Each contribution carries the R-symmetry polynomial Y 11 (σ, τ ) in its respective channel.This is the only so( 8) representation that appears in a given channel, because we are taking the difference of the four-point functions at the two fixed points.Combining this global symmetry structure with the explicit result of [19], we find where the D-function is defined by the integral and we have used the OPE coefficients (4.19).Each line of (4.30) represents a different OPE channel.In writing this, we have used the obvious transformation properties of σ and τ under permutation of the indices, together with (4.21). 14 Eq.(4.30) is the complete result.It is useful to project δF p into a single OPE channelsay, the 12 → 34 channel -by putting it in the form (4.27).This makes it straightforward to extract anomalous dimensions for the double-trace operators [O p O p ].To do so, we project (4.30) onto each representation (ab).Let us define a normalized projection operator P ab|cd (p), that projects Y cd polynomials in the t-channel onto Y ab polynomials in the s-channel: where N ab is the norm, The u-channel projection is identical up to a (−1) a+b .Applied to (4.30), this projector acts on the (σ, τ )-dependent parts of the second two lines, with (cd) = ( 11): one finds The factor of (−1) a+b indicates whether the (ab) representation appears in the symmetric 14 The first line of (4.30) is Y 11 (σ, τ )/2 times the result one would obtain for the same double-trace flow without global symmetries (likewise for the other two channels).The factor of 1/2 comes from contracting the vectors Y 1,2,3,4 with the tensor structure δ IK δ JL + δ IL δ JK − δ IJ δ KL /4 that appears in the two-point function of the Hubbard-Stratanovich field for O 2 , and using the normalization (4.28).See [19] for details.

Double-trace anomalous dimensions
The first line of (4.34) represents the exchange of O 2 , while the second line of (4.34) represents the exchange of double-trace operators of the schematic form The notation indicates that there exist several families of such operators, one for each so (8) representation appearing in the product R p ⊗R p .We denote their total conformal dimension as ∆ n, (p) to be the leading term, of order 1/C T , dual to tree-level contributions to the binding energy in AdS.We focus on the leading-twist operators, with n = 0, and introduce the shorthand γ ≡ γ 0, .
By decomposing the second line of (4.34) into double-trace conformal blocks and working in the 1/C T expansion, we can extract the flow of anomalous dimensions from UV to IR. Define δγ In Appendix B, we carry out the remaining steps in the calculation.The result for even p is where and where is even/odd if a + b is even/odd.(One can easily check that the sum on the RHS of (4.38) equals unity for = 0.) In Appendix B, we also give the result for odd p, and the explicit functions P ab|00 (p) and P ab|11 (p) for a, b ≤ 4 and a + b = even.Note that the ratio of δγ (ab) for two different spins is completely independent of the so( 8) representation (ab).
Actually, we should note that this result for δγ (ab) (p) does not take into account large N operator mixing in ABJM.At leading order in 1/N , the following set of double-trace operators in the ABJM theory have the same conformal dimensions and spins, and therefore undergo mixing: All of these operators must sit in the same so( 8) representation (ab).Therefore, (4.38) should be viewed as a linear combination of contributions from all such operators to the intermediate channel of O p O p O p O p , weighted by their squared OPE coefficients.The full mixing problem is complicated, and we do not solve it here. 15For application in the next subsection, note the simplifying feature that, due to the structure of so(8) tensor products in (4.24), there is no mixing when a = p − 1, p: n, operator can sit in these representations.
Let us make a comment on signs.By inspection, c k (p) > 0 for all p ≥ 4, so the sign of δγ (ab) (p) is given by the sign of the ratio of projectors.In general, these ratios need not be sign-definite: whereas P ab|00 (p) > 0 due to unitarity of mean field theory (see (B.10)), there is no unitarity constraint on P ab|cd (p) for (cd) = (00).Explicit calculation using the projectors in (B.6)-(B.7)does in fact produce both signs for different representations at fixed p. 16 For instance, in the case p = 4, for representations appearing in the symmetric product This pattern appears to generalize to p = 4: the only symmetric representations for which δγ (ab) < 0 are those with a = p, b < p.It would be nice to prove this.

IR dimensions for UV-protected operators
In the tensor product R p ⊗ R p , given in (4.24), the operators living in representations (ab) with a = p − 1, p are protected by SUSY in the UV; all others are unprotected [77].For this subset of protected representations, our double-trace data is especially interesting: since γ n, (p) UV = 0, the change in anomalous dimension under RG flow equals the IR anomalous dimension.
There are further constraints from osp(8|4) representation theory on which composites are protected.In Table 2 we show the relation between internal and spacetime quantum numbers for the protected so(8) multiplets in the (ab) representations.Because ∆ (ab) n, (p) = p + 2n + for protected representations, Table 2 implies the following: The two classes of BPS multiplets of so( 8), specified to the (ab) ≡ [0(a − b)(2b)0] representations, along with the twist τ O = ∆ O − O and Lorentz spin O of the superconformal primary O. See e.g.[36] or Table 2 of [44] for further refinement.
• When a = p, the protected superconformal primary double-trace operators lie in the B series of BPS representations, with = 0.
• When a = p − 1, the protected superconformal primary double-trace operators lie in the A series of BPS representations, with b − = even.
• Among superconformal primaries, only the n = 0 operators are protected. 17 As explained below (4.41), there is no operator mixing for these representations.Thus, equations ( In the UV, these operators are superconformal primaries living in a 1/2-BPS B series multiplet, with vanishing anomalous dimension.In the IR, the anomalous dimension γ (pp) 0 (p), as defined in (4.36), is given in (4.40).By inspection of the projectors through p = 18, we find that for this representation, the ratio of projectors appearing in (4.40) is actually p-independent: This leads to a particularly simple result for the leading-order IR anomalous dimension of 17 These operators have conformal primary descendants, which are also UV-protected; these can be easily enumerated by expanding the supermultiplet operator content.Such conformal primaries may have 0 ≤ n ≤ 3, depending on how many supercharges generate the full multiplet.For n > 0, there is mixing among double-trace operators of n ≤ n and identical spins. in the symmetric (3b) and (4b) representations: where ∈ 2Z ≥0 .These results for γ (ab) (p) at the IR fixed point are the first analytic computations of anomalous dimensions of finite-spin double-trace operators in any large N CFT 3 with an Einstein gravity dual.The only previously known data at finite spin, either analytic or numeric, is a numerical bootstrap estimate in N = 8 ABJM for the so(8)-singlet operators [O 2 O 2 ] 0, for = 0, 2 [44].(Large spin results have been obtained using the lightcone bootstrap [80][81][82][83][84].)This technique could also be applied to derive anomalous dimensions in IR CFTs obtained by double-trace RG flows that preserve a fraction of the UV SUSY: again, certain UV-protected double-trace operators become unprotected in the IR, as determined by the branching rules of the UV superalgebra.The interpretation of CFT anomalous dimensions as AdS binding energies has been discussed elsewhere [85][86][87].

Final comments
The proposal we have made, and the specific example involving ABJM, provide a way to construct non-SUSY CFTs with large N and a large gap that appear to obey all necessary CFT consistency conditions.Of course, it is paramount to understand if there is nevertheless an obstruction.It would be enlightening, though challenging, to fully determine the fate of the moduli space of vacua after the RG flow.If instabilities do develop, the conjecture of [5,6] will have passed a novel test; if they do not, a plausible modification of the conjecture is that all non-SUSY CFTs with a large gap are obtained by SUSY-breaking RG flows.This is still a radical statement that, if true, would be fascinating from the CFT perspective: in the absence of SUSY, the typical large N , large gap CFT is believed to be complicated, with a highly disordered set of irrational operator dimensions and OPE coefficients and a host of possible sporadic phenomena.On the other hand, CFTs constructed via double-trace flow are highly ordered.
We have only computed a handful of gauge-invariant observables of the putative IR fixed point obtained by flowing from ABJM, but it is worth exploring its properties further.For instance, one would like to compute the leading-order shift of the single-trace spectrum in the IR, where O p may acquire anomalous dimensions.We may do so by an AdS computation of the one-loop correction to the propagators of KK modes φ p on AdS 4 ×S 7 /Z k .The relevant bubble and tadpole diagrams -see Figure 2 -can be computed as explained e.g. in [88].More precisely, to compute the IR dimensions of O p , it would be sufficient to compute differences of such diagrams with ∆ = 1 and ∆ = 2 boundary conditions on φ 2 .A missing ingredient are the quartic couplings φ 2 2 φ 2 p , which have never been computed.These would also allow computation of the four-point functions in the large N ABJM theory itself, which remains an outstanding problem. 18n intriguing question, independent of the concerns of this paper, is whether the RG flow (3.4) survives all the way down to small values of N .Can one reach an analog of the Wilson-Fisher model, endowed with so(8) global symmetry, by RG flow from ABJM? Away from large N , the notion of "double-trace flow" is meaningless, but (3.4) can be understood as a fancier version of the typical φ 4 deformation, in analogy with the usual construction of the Wilson-Fisher fixed point via RG flow from the free scalar theory.Such a CFT, if it exists, may (but need not) be a non-SUSY Chern-Simons-bifundamental matter theory.Recent studies of non-SUSY Chern-Simons-matter theories have revealed a rich landscape of fixed points and dualities (e.g.[90][91][92][93][94]); it would be interesting to ask whether this landscape accommodates the so(8) Wilson-Fisher-type theory described above.One promising approach to this problem may be to ask the conformal bootstrap whether such a theory is allowed to exist, for instance, by generalizing the analysis of [44] to include so(8) global symmetry but not SUSY. 19ne might also try to construct SUSY-breaking double-trace flows from large-gap SCFTs in d = 2.It behooves us to look for more M-theory examples.A canonical one is M-theory on AdS 3 × S 2 × CY 3 , whose dual is the MSW CFT with N = (0, 4) SUSY [95].This theory remains poorly understood, but the BPS spectrum is known [96][97][98], and contains no ∆ < 1 operators.It would be worthwhile to seek other examples, particularly given the paucity of explicit constructions of large N CFTs in d = 2 with sparse spectra.We now perturb the CFT by O 2 .To leading order in 1/N , in the IR we take O → σ inside correlation functions, whereupon we must compute the "triangle diagram" with two σ legs; see Figure 3. σ has a two-point function [39,99] The necessary integral is where . . .denotes higher orders in 1/N , and we have used conformal symmetry to go from the first to the second line.Two applications of the conformal integral where where and This projects a t-channel exchange in the (cd) representation of so(8) of the correlator O p O p O p O p , where ∆ p = p/2, onto the s-channel.The identical formula, up to an overall (−1) a+b , holds for projection of a u-channel term onto the s-channel.A nice exposition of the polynomials Y ab (σ, τ ), and a list of those with a ≤ 3, is given in Appendix B of [76].For what follows, we will also need which can be derived from [76].
The projector with (cd) = (00) is relevant for the conformal block decomposition of O p O p O p O p in mean field theory, which is a sum over channels of identity exchange: The projector with (cd) = (11) For both (cd) = (00) and (11), one check on these functions are the zeroes at p = 1, 2, 3: these reflect, correctly, the absence of the (ab) representations in the product R p ⊗ R p for p < a.Note the universal behavior of these projectors at large p, where ∼ 1/p 6 ; in particular, their ratio goes to a constant.The above data are sufficient, using (ab) 0, , for all representations appearing in the symmetric product [R p ⊗ R p ] sym with a ≤ 4. For p = 4, this is the full set of symmetric representations; moreover, for p = 4 and a = 3, 4, these operators are protected in the UV, so δγ (ab) (4) equals the IR anomalous dimensions, as explained in Section 4.3.1.

B.2 Computing δγ (ab) (p)
The starting point for this calculation is (4.34).The strategy is to decompose it into conformal blocks, picking off the terms that contain the anomalous dimensions and applying the results of [19].
The second line of (4.34) contains the double-trace exchanges, which have twist τ = p+2n at infinite N .This can be inferred from the leading power of u: at u 1, a conformal family whose primary has twist τ contributes terms of order u τ /2 times positive integer powers.Each power of the anomalous dimension comes with a power of log u.Putting these facts together, For (ab) in the symmetric/anti-symmetric product of R p ⊗ R p , only even/odd double-trace operators are exchanged.
Having computed the MFT result, we return to (B.8).We focus on the leading twist operators, with n = 0, henceforth, and use the shorthand γ ≡ γ 0, and a 0, = a .This allows us to utilize the lightcone limit, u 1 , v fixed .(B.12) In this limit, g p, (u  C Contrast with SUSY-breaking N = 4 SYM orbifolds For context, we give a brief historical overview of one of the most well-studied non-SUSY constructions, namely, the AdS 5 × S 5 /Γ orbifolds, dual to orbifolds of 4d N = 4 SYM [101,102].Non-freely-acting orbifolds Γ = Z k have AdS tachyons; explicit CFT calculations at weak coupling revealed the existence of an unstable effective potential [33,61], and, later, nonzero beta functions for double-trace operators comprised of twisted sector single-trace operators [49,50].The AdS tachyons were conjectured to be the strongly coupled avatars of these weak coupling phenomena [33].The freely-acting orbifolds have no tachyons in AdS, but do suffer from a non-perturbative instability [34]; though an initial weak coupling calculation [33] revealed no apparent instability of the effective potential, these theories were nevertheless shown to break conformality [49,50].The field theory picture was tied together by [51], who showed that along any fixed line of d = 4 CFTs with adjoint fields, and at any value of the marginal coupling, the CFT develops nonzero beta functions if and only if the perturbative vacuum at the origin of moduli space is unstable.No such examples survived further scrutiny.The orbifold studies were extended in [13] to AdS 4 × S 7 /Γ, and to the "skew-whiffed" AdS 4 × S 7 background [10], where global singlet marginal operators were found in both cases; this is likely to imply a breaking of conformality, but deserves further study.
Let us also highlight the "orientifold" CFT construction of [14], which does not develop nonzero double-trace beta functions [103], and is not known to suffer from other instabilities.It should be noted, though, that this theory ceases to be conformal away from strictly infinite N , so it does not belong to a sequence of CFTs with a large N , large gap limit.

. 1 )Figure 1 :
Figure1: A cartoon of a general double-trace flow.In this paper, we take the UV theory to be a superconformal field theory with large N and a large gap, and O to be a superconformal primary.

Figure 2 :
Figure 2: The bulk diagrams needed to determine the leading anomalous dimensions of the single-trace operators O p after the ABJM double-trace flow by O 2 2 .

Figure 3 :
Figure 3: The triangle diagram determines the three-point coupling Φσσ in the IR, to which the UV coupling ΦOO flows.The purple points are integrated over.
).In the dual AdS theory, this corresponds to choosing ∆ + boundary conditions for all components of the m 2 = −2 scalar field dual to O. Our proposal is that such IR fixed points may be viable CFTs for finite k.More precisely, this statement is on firmest footing for k = 1, 3, as we explain below.
fact that O 4 O 2 O 2 IR = O 2 O 2 O 2 IR = 0 after flowing from ABJM modifies the leading large-N behavior of thermal one-point functions in the IR.Consider first the one-point function of O 4 , defined on S 1 β × S 2 as O 4 S 1 β ×S 2 = Tr H (O 4 e −βH ) (4.8) where H is the local operator Hilbert space.The leading low-temperature asymptotics are determined by the dimension of the lightest operator to which O 4 couples linearly.Thanks to the result above, the thermal one-point function of O 4 has different behavior in UV and IR: UV : O 4 S 1 β ×S 2 ≈ O 2 O 4 O 2 e −β + . . .IR : O 4 S 1 β ×S 2 ≈ O 4 O 4 O 4 e −2β + . . .
16)We will consider four-point functions O p O p O p O p for p > 2, and compute their change under the RG flow triggered by(3.4): that is, under the deformation (3.4), we compute the quantityO p (x 1 ; Y 1 )O p (x 2 ; Y 2 )O p (x 3 ; Y 3 )O p (x 4 ; Y 4 ) IR − O p (x 1 ; Y 1 )O p (x 2 ; Y 2 )O p (x 3 ; Y 3 )O p (x 4 ; Y 4 ) U V(4.17) to leading order.This is controlled, roughly speaking, by the order g term in O p O p O p O p e −g O 2 2 O ) -the numerator of the last factor in (4.5) blows up; this gives a finite result only if C U V ΦOO = 0.
A.6)lead to the result (4.5) quoted in the text.One concludes that UV-extremal correlators involving O vanish in the IR.Note that the reverse is also true: if the correlator is not extremal in the UV, but becomes extremal in the IR -that is, if ∆ Φ = 2(d − ∆

5 )
The first several low-lying representations (ab) in the symmetric product [R p ⊗ R p ] sym are was needed for the conformal block decomposition of the change in O p O p O p O p after double-trace flow triggered by O 2 2 , where O 2 sits in the (11) representation.The first several low-lying representations (ab) in the symmetric product [R p ⊗ R p ] sym are [19] is essentially identical to the same problem in the case where O p is uncharged under global symmetry -in particular, it is clear that δγ (ab) (p) is simply the uncharged result, multiplied by the ratio of projectors.The uncharged problem was solved in[19], using the fact that g coll (v) obeys a simple orthogonality condition.The result is /odd if a + b is even/odd.This result is valid for any p ∈ Z >0 .Moreover, it turns out that for p ∈ 2Z and ∈ Z ≥0 , this function simplifies tremendously: