Double-trace deformations of conformal correlations

Large N conformal field theories often admit unitary renormalization group flows triggered by double-trace deformations. We compute the change in scalar four-point functions under double-trace flow, to leading order in 1/N. This has a simple dual in AdS, where the flow is implemented by a change of boundary conditions, and provides a physical interpretation of single-valued conformal partial waves. We extract the change in the conformal dimensions and three-point coefficients of infinite families of double-trace composite operators. Some of these quantities are found to be sign-definite under double-trace flow. As an application, we derive anomalous dimensions of spinning double-trace operators comprised of non-singlet constituents in the O(N) vector model.

have long been part of the vector model paradigm, in which the critical boson and fermion theories [1,2] may be reached by double-trace deformations of free bosons and fermions, respectively. They also have a natural interpretation in the context of AdS/CFT, where they are generated by a change in boundary conditions [3][4][5].
The effect of such deformations at the level of the CFT partition function [6], and the relation to AdS boundary conditions [3,4,[6][7][8][9][10][11], was clearly laid out in the early days of AdS/CFT. In this work, inspired by a contemporary perspective on the value of correlation functions to the foundations of CFT, we compute some more complicated quantities: the change in CFT four-point functions under double-trace flow from UV to IR, and the effect of such flows on the sector of double-trace operators in the IR CFT. The calculations have simple bulk duals in AdS, an elegant relation to harmonic functions for the conformal group, and reveal new observables whose change from UV to IR is sign-definite.
A double-trace flow is defined as follows. Consider a CFT d that admits a large N expansion, and possesses a single-trace scalar operator O with ∆ < d/2. Then we can consider the relevant double-trace deformation where the subtraction enforces the primary condition. There is one operator for each (n, ), with conformal dimensions ∆ n, = 2∆ Φ + 2n + + γ n, (1.3) for some γ n, . We now know that their quantum numbers are a rich source of dynamical information about interacting large N CFTs: they are sensitive to the coupling strength; they are constrained by causality and unitarity [13][14][15]; crossing symmetry of four-point functions relates the exchange of double-trace operators to Regge trajectories of singletrace operators [16][17][18], and hence γ n, to the cusp anomalous dimension and the higher spin gap scale; and in the AdS/CFT context, the nature of inter-particle forces in the bulk is manifest in the behavior of γ n, as a function of (n, ), thus making γ n, a sensitive probe of bulk locality [19][20][21][22][23][24][25][26]. These reasons motivate the study of how the γ n, change under double-trace flow. The γ n, are 1/N -suppressed, so it would seem more difficult to compute their change under double-trace flow. The same is true for three-point functions ΦΦ[ΦΦ] n, . However, the change of the γ n, from UV to IR can be extracted from the leading-order change in the connected part of the four-point funtion ΦΦΦΦ , which itself may be straightforwardly computed. 1 In section 2, we begin with an exercise, in which we compute the change in threepoint functions OOO , ΦOO and ΦΦO . These results are known -for instance, by modifying the boundary condition in AdS calculations of CFT three-point functions -but will appear in our later analysis and serve as a useful warmup.
In section 3, we compute the change in connected four-point functions ΦΦΦΦ and ΦΨΦΨ from UV to IR, where Φ and Ψ are distinct scalar primaries. Part of our message is that the result, which is manifestly crossing-symmetric, may be expressed simply in terms of a singleD-function -see (3.9) and (3.31). TheD-functions are themselves not elementary, but arise in many contexts in CFT at both weak and strong coupling (e.g. [27][28][29][30][31]). In section 3.2, we review the manifest equivalence [10,32] between our CFT calculation and a tree-level AdS calculation. Moreover, the change in these four-point functions is technically essentially identical to the computation of four-point conformal partial waves for principal series representations, whose utility has recently been emphasized [33][34][35][36][37][38]. 2 Thus, our result may be viewed as a novel physical interpretation of these objects in terms of doubletrace flows, and provides for them a mathematical expression in terms of aD-function.
In section 4, we use the change in four-point functions to extract the change in OPE data of the leading-twist double-trace operators [ΦΦ] 0, and [ΦΨ] 0, , for all . We focus mostly on anomalous dimensions γ ≡ γ 0, , although we compute some OPE coefficients as well. The results for δγ ≡ γ IR − γ UV , in (4.20) and (4.32), have various universal features. We wish to highlight one here: for [ΦΦ] 0, operators, δγ 0 as given in (4.11) is independent of ∆ Φ . It is also always positive for unitary values of the dimensions (∆ Φ , ∆). Therefore, we arrive at the interesting conclusion that this privileged class of RG flows admits sign-definite quantities over and above the usual a/c/F observables.
In section 5, we specify the conformal dimensions ∆ and ∆ Φ to certain values where the results simplify, and use these results as a tool to derive anomalous dimensions of some double-trace operators in the O(N ) vector model, in which the constituents are not O(N ) singlets. This is possible because δγ = γ IR , since the UV theory from which the O(N ) model descends is free. Specifically, we derive ΦΦ * ΦΦ * where Φ and Φ * are conjugate operators in the O(N ) model, each a scalar bilinear in the rank-two symmetric traceless representation of O(N ), and extract γ for double-trace operators [ΦΦ * ] 0, . The result for γ in various spacetime dimensions can be found in (5.7). This is, to our knowledge, a new result. It exhibits interesting harmonic behavior in d = 4 − . While our technique here -of using a calculation of δγ between fixed points to derive γ at the IR fixed pointis essentially identical to an ordinary large N calculation in the O(N ) model, it would be interesting to apply it to other CFTs beyond the O(N ) model, where it acts as a simpler alternative to standard computation of a full four-point function. A recent application can be found in [47].
Appendices A-D contain some background material, various details of calculations in the text, and a conformal perturbation theory cross-check of our result for γ IR 0 in the O(N ) model in d = 4 − .

Warmup: three-point functions
To introduce some formalism, and to derive a result we will use later, let us first compute the change of some three-point coefficients. All of our calculations will rely on the Hubbard-Stratonovich auxiliary field method, which we recall now.
Upon introducing the deformation (1.1), the large N expansion in the IR can be developed by introducing an auxiliary field as where the quadratic term −σ 2 /(4λ) can be dropped in the IR limit. The auxiliary field σ acquires the following induced two-point function at leading order in 1/N (see e.g. [48,49] for a review and more details) Thus in the IR limit σ, which replaces the operator O, becomes a scalar primary of dimension d − ∆ + O(1/N ); that is, O turns into its "shadow". All other single-trace operators have the same dimension in the UV and IR to leading order at large N . Correlation functions at the IR fixed point can be computed systematically in the 1/N expansion using the σ propagator (2.2) and the σO vertex in (2.1). 3 Note that (2.1) can be thought as a kind of Legendre transform relating UV and IR correlators.
To compute the change in three-point functions, we first note that all three-point functions of single-trace operators that do not involve the perturbing operator O(x) are unchanged to leading order at large N . This is transparent in the AdS picture, as the corresponding tree-level three-point Witten diagrams that do not involve O(x) are unaffected by the change in boundary conditions.
For what follows, we will define norm-invariant squared OPE coefficients Let us first consider the OPE coefficient C ΦΦO , where Φ is a single-trace scalar operator other than O. Conformal three-point functions take the form In the IR we replace O by σ, giving the three-point function where . . . denotes subleading orders in 1/N . See figure 2. To leading order at large N , we evaluate this expression using the three-point conformal integral (e.g. [30,50]), Specialized to our case, we thus obtain 3 One has to be careful not to include one-loop bubble corrections to σ lines, as those are already resummed when using the effective propagator (2.2). The factor in parenthesis is the OPE coefficient C ΦΦσ . Using the normalized squared OPE coefficients introduced in (2.4),

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the above calculation gives This can be seen to match a previous derivation using AdS integrals [51]. Notice that ∆ ↔ d − ∆ properly swaps the labels UV and IR. Similarly, we can compute the change from UV to IR of the C ΦOO OPE coefficient by attaching two σ lines to the UV three-point function ΦOO . This was worked out explicitly in [47], following similar steps as described above. In terms of normalized squared OPE coefficients, the result is Finally, the three-point function OOO in the IR can be computed by attaching three σ lines to the UV three-point function and using repeatedly the conformal integral (2.7). This yields Note that for d = 3 and ∆ = 1, a IR OOO vanishes, which is a well-known result for the critical O(N ) model in d = 3 [52].

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will compute the change in the connected four-point functions involving Φ and Ψ under the renormalization group flow triggered by δS CFT = d d x O 2 , with ∆ < d/2. In what follows, we label the difference between UV and IR observables X as

Identical operators
For the four-point function ΦΦΦΦ , conformal symmetry constrains this difference to take the form is a function of the conformal cross-ratios that is undetermined by conformal symmetry. Our goal is to determine δF . The connected four-point function of Φ in the IR can be computed in the large N expansion as To leading order at large N , we can factorize the six-point function in the second line as where the permutations account for the t-and u-channels. So the problem is essentially just to compute the "two-triangle" diagram with a σ field exchange, where each triangle is the three-point function ΦΦO in the UV CFT. See figure 3. From the AdS point of view, this computes the difference of two four-point exchange Witten diagrams with external Φ legs and exchange of the bulk field dual to O, after taking the difference of boundary conditions on the exchanged field. We demonstrate that explicitly in section 3.2.
Using the form of the conformal three-point function in (2.5), the "two-triangle" diagram is given by The integration can again be performed using conformal integrals. First, we integrate in z 1 using the three-point integral (2.7). Next, we integrate over z 2 using the four-point JHEP02(2018)175 conformal integral [53] The functionD ∆ 1 ∆ 2 ∆ 3 ∆ 4 (u, v) defined above is the ubiquitousD-function that appears in calculations of AdS Witten diagrams, see e.g. [30] and appendix B. Note, however, that thē D-functions appearing here are of a special type: the sum of the exponents is equal to d. After these steps, equation (3.6) yields Note that this takes the expected conformal form. Adding all terms related by exchanging the external points, we arrive at the final, crossing-symmetric result. In terms of the normalized OPE coefficient a UV ΦΦO defined in (2.10), and using the definition of C σ in (2.2), we can write this in the form (3.2) with This is one of our main results. The change in the connected part of four-point functions under double-trace flow takes a very simple form, expressed as a manifestly crossingsymmetric sum ofD-functions, one from each channel. We can equivalently write this in terms of a IR ΦΦO using (2.11).
upon identifying Φ with φ i , and O with σ ∼ φ i φ i . This is because, even though φ i is not strictly speaking a "single-trace" operator, the calculation of the leading large N contribution to the connected correlator takes the same form as the two-triangle diagram considered above, with the role of the triangle played by the three-point functions in the UV free theory. To account for the index structure we just need to slightly generalize our calculation: adding up the singlet contributions in each channel and usingD-function identities summarized in appendix B, the expression in brackets in (3.9) may be written This is indeed the correct result for the connected correlator in the critical O(N ) model [54]. 4

Equivalence between CFT and AdS calculations
There is a manifest equivalence between the CFT result in the above language, and the dual AdS calculation, as previously discussed in [10,55]. The crucial fact in proving this equivalence is the following identity for the difference of bulk-to-bulk propagators with standard and alternate boundary conditions, shown pictorially in figure 4: (3.12) C σ was defined earlier as the normalization of the σ propagator, eq. (2.2). Here and elsewhere in this subsection we use the notation x ≡ (z, x), y ≡ (w, y) to denote points in AdS d+1 in the usual Poincare coordinates, with x, y etc. denoting coordinates of the flat boundary at z = 0. In this identity G ∆ (z, x; w, y) is the bulk-to-bulk propagator with its canonical normalization, i.e. satisfying 13) and the bulk-to-boundary propagator is normalized as Let us recall that with this choice of normalization of the bulk-to-boundary propagator, the two-point function of the dual operator is normalized as JHEP02(2018)175 It is straightforward to prove (3.12) by assembling some known ingredients. First, one starts with the following single-integral identity [10,56]: Next, we use the following relation between the bulk-to-boundary propagators of conjugate dimension, which can be obtained by straightforward integration From this formula we see that changing the boundary conditions on bulk-to-boundary lines essentially amounts, from CFT point of view, to attaching the propagator of the auxiliary field σ. We can use (3.16) to rewrite (3.15) in the form (3.12); we have used the fact that in this AdS calculation we can identify C OO = C ∆ . We make some further remarks on the relation of these identities to AdS harmonic functions in appendix A. Given (3.12), the relation to the CFT calculation of the "two-triangle" diagram is transparent. The difference between the two exchange Witten diagrams in the channel 12 → 34 for different boundary conditions on the intermediate field is, using (3.12), where λ ΦΦO is the AdS cubic vertex. The AdS integrals in brackets, each involving three bulk-to-boundary propagators, just give the three-point functions in the UV: in a normalization where C ΦΦ = C ∆ Φ , and C OO = C ∆ . Thus, we see that (3.17) precisely reproduces the "two-triangle" diagram (3.6) in the corresponding channel. Let us emphasize that our calculation does not rely on strong coupling, rather, only on large N factorization and conformal symmetry.

Relation to SO(d + 1, 1) harmonic analysis
The result (3.8) for the difference in four-point functions in a single channel should equal a difference of conformal blocks for O exchange in that same channel. This is clear from the AdS perspective explained in the previous subsection, combined with the fact that, in the direct channel conformal block decomposition of an AdS exchange diagram, the double-trace OPE data depends only on the squared mass, not the quantization, of the exchanged operator (e.g. [57]). Specifically, accounting for the difference in UV and IR OPE coefficients, we want to check that where G ∆,J (u, v) is the conformal block for exchange of a dimension-∆, spin-J operator. Note first that neither side depends on ∆ Φ . For the conformal blocks, we use the series representation [30], . For theD function, we also use the series representation [30], which we can write as Accounting for the ratio a IR ΦΦO /a UV ΦΦO given in (2.11), one finds agreement term-wise. Now, conformally-covariant four-point functions may be expanded in a complete basis of single-valued functions living in the principal series representations of the conformal group, with ∆ = d 2 + iν, where ν ≥ 0 and is real (see e.g. [33][34][35][36][37][38] and references therein). In the notation of [38], let us call these single-valued functions Ψ ∆,J (x i ). These functions are not the ordinary conformal partial waves for unitary representations ∆ ≥ d − 2 + J with ∆ ∈ R, which are not single-valued. The Ψ ∆,J (x i ) may be written in terms of the conformal blocks as Now, we note that where a IR ΦΦO /a UV ΦΦO was computed in (2.11). Therefore, from (3.19), we may write Ψ ∆,0 (x i ) simply in terms of a singleD-function: This is a concise expression, valid for any d. We are giving both a mathematical expression for Ψ ∆,0 (x i ), as well as a physical interpretation in terms of double-trace RG flows. In AdS language, Ψ ∆,J (x i ) is, up to a prefactor, equal to the difference of exchange diagrams for which the exchanged operator takes ∆ + and ∆ − quantizations. That this is true (for any J) is obvious from the identities of the previous subsection and appendix A: CFT harmonic functions are dual to AdS harmonic functions.

Generalization to pairwise identical operators
The CFT calculation of the two-triangle diagram can be straightforwardly generalized to the case of four different external operators. With an eye toward applications, we explicitly consider here the case of pairwise identical scalar operators: given two scalar primary operators Φ and Ψ, we may compute the difference in the connected four-point function ΦΨΦΨ between UV and IR fixed points, Defining the norm-invariant ratio

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In the event that ΦΦO = 0 or ΨΨO = 0 -for instance, if O is neutral under a global symmetry under which Φ and Ψ are charged -the full, crossing-symmetric result is a sum of two terms, 5 one in each available channel: Note that this is symmetric under ∆ Φ ↔ ∆ Ψ , as it must be. This follows from theDfunction relations in appendix B. As in section 2, one can also derive the ratio between UV and IR three-point coefficients; the result is In these cases, we have zeroes from the gamma functions in (3.31) for all p ∈ Z ≥0 . Consequently, the result simplifies even further, which we now show for p = 0. (See appendix C.2 for the p = 1 result and comments on general p.) ) First, note that the three-point coupling (3.32) vanishes in the IR, due to the Γ −2 (0) factor: This happens because for ∆ + ∆ Ψ = ∆ Φ -indeed, for ∆ + ∆ Ψ = ∆ Φ − 2p for p ∈ Z ≥0the UV three-point function is "extremal" [45]. As explained in [47], changing boundary conditions on an operator in an extremal correlator implies its vanishing at the new fixed point. Likewise, the prefactor in (3.31) vanishes. In order to extract the result for δF , we must introduce a regulator. Take ∆ + ∆ Ψ = ∆ Φ + 2 . Then To determine the finite piece of (3.31), we must extract the O( −2 ) term in theD-functions, to cancel the 2 prefactor. Using their double-sum representation (see appendix B), one simply finds 6D 5 See appendix C.1 for the most general result in which these OPE coefficients are nonzero; there is simply one more term, of the same functional form as (3.29). 6 In particular, everything other than the Γ 2 ( ) is regular when → 0, including the G-functions (B.8).

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Combining this with (3.31), we get This is the final, extremely simple, result. Remarkably, (3.36) is d-independent, and is manifestly negative for all real u, v. We will soon see other sign-definite properties of the double-trace RG flow. We focus for now on the result (3.9) for the case of identical external operators. In general, the four-point function has the conformal block expansion  n, are known in general d [58]. This, in turn, induces a 1/N expansion of F (u, v) at each fixed point. Taking the difference of IR and UV connected correlators to leading order in 1/N , Recall that we define N as cT ∼ N .

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and n, δγ n, ∂ n + δa Writing (3.9) in the form (4.3)-(4.5) and expanding in powers of u and 1−v, we can extract all OPE data. In section 3.3, we showed that the O exchange piece δF O (u, v) is indeed accounted for by the first term in (3.9), i.e. the direct-channel term. In the rest of this section, we focus on the double-trace terms (4.5), which come from crossed-channel contributions and contain interesting data. 8

Double-trace anomalous dimensions
First we extract δγ n, , focusing in particular on the leading-twist tower n = 0. (Higher n may be computed in a systematic expansion in small u.) The final result is in (4.20). In this subsection and elsewhere, whenever we focus on the n = 0 tower for some double-trace observable X, we use the notations We proceed by isolating the log u piece of the full double-trace contribution In what follows, we will concentrate on δγ . So to extract the anomalous dimensions, we will need the expansion of the log u terms ofD-functions in (3.9) in the OPE limit. These terms are given by 9 where the G-function, introduced by Dolan and Osborn [50], admits the following doubleseries expansion: Note that the G-function obeys a small-u expansion 8 A complete way to perform these calculations is to use Caron-Huot's inversion formula [36]. This requires taking a double-discontinuity of ourD-function. We will instead use more basic tools. 9 Note that theD d−∆ JHEP02(2018)175 Before giving a general result for δγ , let us start by extracting δγ 0 , the change in the anomalous dimension of the leading-twist scalar operator : Φ 2 : = [ΦΦ] 0,0 . In this case, we have a τ, = a 2∆ Φ ,0 = 2 + O(1/N ), and to leading order at small u we just have Then, using (4.8) into (3.9) and matching to (4.1), we find the result (4.11) We point out two features of this result. First, it is manifestly positive for all d−2 2 < ∆ < d 2 . Second, it is highly non-trivial that this does not depend on ∆ Φ , because γ IR 0 and γ UV 0 both do. This ∆ Φ -independence will not persist at higher . Note that, using (2.11), we can also express (4.11) as which is manifestly odd under ∆ → d − ∆, as required.

Leading-twist
We now derive δγ in closed form. First, we introduce a notation In the lightcone regime u 1, the conformal blocks become the collinear SL(2, R) blocks, 10 (4.14) To leading order at small u, the log u term of (4.7) becomes with eigenvalue β(β − 1). They obey the orthogonality condition where β − β ∈ Z, and the contour runs counterclockwise around the origin. This was used in a similar context in e.g. [19,26]. Applying this to (4.15),

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which is the desired result. Actually, we may go further and explicitly extract the residue in closed form: from (4.10), it is clear that we need only isolate a term of order x in the product of various hypergeometric functions. We carry this out in appendix C.3. The final result can be written as a finite sum: where a (0) are the mean field theory OPE coefficients [19,58] This can be neatly written in terms of a terminating 4 F 3 hypergeometric function, with no explicit appearance of a (0) : with δγ 0 given in (4.11). As a consistency check on this result, evaluating (4.20) in the large limit, one finds This correctly reproduces the leading-order results of the lightcone bootstrap [20,21]: it is a difference of the leading large-spin asymptotics of δγ in the IR and UV.

Comments
RG monotonicity. As we noted earlier, it is interesting that δγ 0 in (4.11) is always positive under double-trace RG flows: that is, γ 0 is greater in the IR than the UV, for any spacetime dimension d. What about δγ for higher spins? At large spin, ∆ d−∆ , so the UV term of (4.21) dominates (assuming a UV ΦΦO = 0), implying δγ 1 > 0 due to the positivity of a UV ΦΦO . Thus, any negativity must be confined to finite . By a combination of analytical arguments and numerically sampling many values of parameters, we find the following condition: This implies that γ IR ≥ γ UV for d ≥ 4 , where ψ is the digamma function. The prefactor is manifestly positive for all ∆ Φ > 1 and > 0. The difference of digamma functions is also positive, because ψ (x) > 0 for x > 0. Therefore, δγ is indeed positive under double-trace RG flow. For d = 4, one can show that at ∆ = d/2 and ∆ Φ = d/4, δγ has a zero for all ∈ Z, because δγ δγ 0 ∆=d/2, Then by plotting many values, one sees that (4.23) holds. In figure 5, we exhibit this behavior in d = 3. This conclusion would follow if, as suggested by the sampling, ∆ Φ = d/4 is the only zero of δγ at ∆ = d/2, viewed as a function of ∆ Φ , for unitary values of ∆ Φ .
Flows from UV free CFTs. One may be puzzled about cases in which the UV is a free theory, so that δγ = γ IR . In such cases -again assuming a UV ΦΦO = 0 -one has γ IR 1 > 0, because the UV term of (4.21) dominates. This conflicts with naive lightcone bootstrap intuition at large spin. The resolution to this is that the UV free theory contains an infinite tower of conserved higher spin currents which becomes nearly conserved in the IR, and negativity of γ does not apply, even at large spin [17]: a resummation is required. What our result shows is that, in fact, every CFT with slightly broken higher-spin symmetry that is obtained by double-trace flow from a UV-free CFT has γ IR 1 > 0.

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Heavy operators. Consider the result (4.20) for the change, under double-trace flow, of the anomalous dimensions of the double-trace operators [ΦΦ] 0, . We now suppose the external operator Φ is a heavy operator, with 1 ∆ Φ N . Such an operator may be, for instance, a string-scale operator in a large N gauge theory.
First, suppose that remains finite as we dial ∆ Φ 1. In this case, δγ = δγ 0 to leading order in 1/∆ Φ . The reason for this is clear in AdS: the binding energy for a bound state of two heavy particles with mL AdS 1 will be unaffected by the addition of a parametrically small angular momentum J m. More interesting is the regime in which Representing the 4 F 3 in series form and taking the limit of the summand, one obtains a finite result; performing the sum then yields an ordinary hypergeometric function, One readily confirms that for η → ∞ -that is, 1 ∆ Φ -we recover the ∆ Φ 1 limit of the large spin expansion (4.21). It would be interesting to reproduce (4.28) from a bulk computation in which δγ is the difference in the contribution of O, for standard versus alternate quantizations, to the binding energy of the [ΦΦ] 0, state, where Φ is represented as a particle moving along a bulk worldline.

Double-trace OPE coefficients
One can also derive the change in OPE coefficients, δa (1) n, , in (4.5). We eschew a comprehensive treatment here, only giving the lowest-lying contribution. Again starting from (3.9), we use the form of theD-functions in appendix B -in particular, equations (B.6) and (B.10) -to obtain δa (1) where γ E is the Euler constant. Note the interesting feature that, like δγ 0 , this is independent of ∆ Φ . There is simplification for various rational values of ∆, d. One can continute iteratively for low as desired.

Generalization to pairwise identical operators
We now perform the same analysis for the pairwise identical correlator ΦΨΦΨ , whose change under the RG flow was derived in (3.31). Let us start by deriving the change in anomalous dimensions, δγ , for the leading-twist double-trace operators [ΦΨ] 0, . The strategy for the calculation is the same as for the correlator ΦΦΦΦ : starting with the result for δF (u, v) in (3.31), we expand in collinear blocks and apply a projector. Some formulas become rather unwieldy, so we present the results here and describe the detailed calculation in the appendix C.4. We arrive at the following generalization of (4.18):

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Using the explicit values of the infinite N OPE coefficients a (0) from [58], the sum above can be rewritten as where the spin-zero anomalous dimension is These are known to be symmetric in the last four arguments, which includes the transformation ∆ Φ ↔ ∆ Ψ . In parallel with section (4.2), one may also derive, δa (1) 0 , which is now the difference in (normalized) squared OPE coefficients C 2 ΦΨ[ΦΨ] 0,0 ; the result may be found in (C.6).

Extremal case: ∆ Φ = ∆ Ψ + ∆
Recall that for this extremal alignment of dimensions, we obtained the simple result in (3.36), where we assumed that the ΦΦ → O → ΨΨ channel is absent. The first term in (3.36) represents the UV exchange of O, It is not clear whether the orthogonality property is physically interesting here. It is also interesting to note that these and the related "Wilson functions" have appeared recently in the physics context as fusion matrices in 2d CFT and as scattering phases near AdS black holes [33,34,61]. In the present context, the δγ should be thought of as 6j-symbols for the confomal group, which would follow from their derivation (not performed here) from the inversion formula [36]. We thank David Simmons-Duffin for discussion on this and related issues.

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where we have used that a IR ΦΨO = 0 in the first line. The fact that the conformal block is simply equal to unity can be checked explicitly using the d = 4 blocks, When ∆ 12 = ∆ 34 = ±∆, indeed one has g ∆,0 (z, z) = 1. The second term in (3.36) represents the exchanges of [ΨΦ] n, : The absence of a log u term implies that, consistent with (4.33), Moreover, due to the simple form of the result, we can derive explicit formulas for δa (1) .
In terms of conformal blocks, (4.38) comes from a sum, over all n, , due to the change in OPE coefficients δa (1) n, : in particular, they obey the sum rule ∞ n, The right-hand side is independent of u. This implies that if we expand the left-hand side in powers of u, we obtain an infinite set of equations. First, at zeroth order, ∞ =0,2,...
where g coll 0, (u, v) is the n = 0 collinear block for pairwise identical operators in the ΦΨ → ΦΨ channel, defined in (C.16). Since (4.41) is a sum over collinear blocks, it can be solved for δa (1) by using the projector for the collinear blocks, just as for the δγ in appendix C.4.
The result is As a check, this agrees with the (C.6) derived for general ∆ Φ , ∆ Ψ , when specialized to the case ∆ Φ = ∆ Ψ + ∆. Note that at spin-zero, which is much simpler than the spin-zero result for identical operators in (4.29). Moreover, all terms in (4.40) carrying powers of u must vanish. This is allowed by unitarity because there is no sign constraint on the δa where we must permute the indices of T R O 1234 as well as the positions of the operators. 12 Note that, upon decomposing this into a single OPE channel, we must project T onto a crossed channel; in doing so, multiple representations will generically appear, not only R O . Such projections were carried out for a specific example where G = SO(8) in [47], involving a double-trace flow from the ABJM theory [62].

Applications
In this section, we use our results to derive new double-trace data in interacting vector models in various d. We also specialize the operator dimensions to certain values where our results for δγ simplify.

∆ Φ = ∆ = d − 2: vector models
In the special case ∆ Φ = ∆ = d − 2, our results can be used to extract predictions for the four-point functions and corresponding OPE data of certain non-singlet operators in the O(n) vector model, as we now explain.
Let us start with n = N M free scalars ϕ ia , where i = 1, . . . , N, a = 1, . . . , M , and take N to be large with M fixed. This defines a free CFT with O(n) global symmetry, but we 12 We have absorbed the overall coefficient into the definition of T R O 1234 .

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can now look at the spectrum of singlet operators under the O(N ) symmetry rotating the i-index. The single-trace scalar operators of the free CFT in the O(N ) singlet sector are  As a particularly tractable example where we can directly apply our results for pairwise identical operators in sections 3.4 and 4.3, we can consider the case M = 2. We can then introduce a complex basis The two symmetric traceless operators Φ ab in (5.1) correspond to linear combinations of the complex operators from UV to IR is given by (3.31), provided we identify Ψ = Φ * and take ∆ = ∆ Φ = ∆ Ψ = d − 2. 13 The formula for the change in anomalous dimensions of the double-trace operators 13 Note that we have ΦΦ * O = 0 but ΦΦO = 0 due to U(1) charge conservation.

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[ΦΦ * ] 0, is then given by (4.20) with ∆ = ∆ Φ = d − 2, and is valid both for even and odd . In fact, since the UV theory is free, γ UV n, = 0. Therefore, in these cases, and we can use our formulas to read off the anomalous dimensions in the interacting IR CFT. The same observation was made in [47], where γ UV = 0 due to supersymmetry for UV-protected double-trace operators. We will denote γ IR simply by γ below. For various values of the spacetime dimension d we find, from (4.20), where H x = x n=1 1/n is the harmonic number, and where we used a UV = 4/N , which can be found by Wick contractions. From (C.6), we can also find the change in OPE coefficient of the double-trace scalar operator, which is simply Note that the sign is negative for all d > 2. Let us analyze these results. We observe that for all d > 4, the anomalous dimensions γ grow like γ 1 ∼ −2 (5.10) consistent with the lightcone bootstrap [20,21]. This follows from the previous formulas and where γ E is the Euler constant. In d = 3, γ 1 ∼ −1 ; this is also consistent, because the tower of slightly broken higher spin currents with τ = 1 + O(1/N ) furnishes the leadingtwist sector of the Φ × Φ * OPE instead of O.
The case of d = 4 is somewhat special: we really should work in d = 4 − , since (to leading order in 1/N ) We may also write (5.13) in terms of the conformal spin, 14 At large J, after the log J/J 2 term, the expansion is in even powers of J −2 [66]. It is interesting to note the similarity of our result to the one obtained in section 4 of [67], where φ 2 φ 2 φ 2 φ 2 was computed in a small deformation of a free scalar CFT in d = 4. On general grounds, the first-order anomalous dimensions of the "single-trace" currents, J = φ∂ φ were found to be γ(J ) = c 1 + c 2 H 1 , where now J 2 = ( + 1), and the c i are constants that could not be fixed by symmetries alone. It would be interesting to reproduce our results (5.7) using slightly broken higher spin symmetry, which may give a natural explanation for the appearance of harmonic functions.
The anomalous dimensions (5.12)-(5.13) may be also computed directly by conformal perturbation theory methods in d = 4 − , at finite N ; as a check of our results, we outline this calculation in appendix D for the case = 0. The final result is which in turn can be seen to match our prediction (5.12) at large N . Note that in our notation, ∆ ΦΦ * = ∆ projectors as explained in [47]. In fact, since the exchanged operator O(x) is an O(M ) singlet, the tensor in (4.45) is trivial, and role of the projectors simply cancels out when expanding in a given OPE channel. Hence, the resulting anomalous dimensions are the same as those listed above in (5.7).

∆
In this case, computing the change in anomalous dimensions of [ΦΦ] 0, operators in the IR, the result (4.20) for low values of d is where we expanded δγ 0 explicitly near d = 4 using (4.11).
One class of UV CFTs in d = 3 in this category comes from supergravity compactifications on AdS 4 × M 7 , in which M 7 has non-trivial internal cycles. In particular, for any Sasaki-Einstein M 7 with a nonzero second Chern number b 2 , the CFT has an extra N = 2 conserved current multiplet, the Betti multiplet, due to wrapped M2-branes. These multiplets contain ∆ = 1 and ∆ = 2 scalars that are singlets under all global symmetries, which we identify with O and Φ, respectively, in the calculation above. The ∆ = 1 Betti scalar is parity odd, so the CFT must break parity in order that a UV ΦΦO = 0. A parity-breaking mechanism using internal fluxes for CFTs with AdS 4 ×M 7 duals was introduced in the context of the ABJ theory [70], where M 7 = S 7 /Z k , and applied to other M 7 with b 2 = 0 in e.g. [71].

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A AdS harmonics and propagator identities The identity (3.15) is closely related to the so-called "split", or harmonic, representation of the bulk-to-bulk propagator (see e.g. [39,41,72]). Since the bulk-to-bulk propagators with either boundary condition satisfy the same equation (3.13), their difference must be proportional to an AdS harmonic function (see e.g. [73] for a review), which may be defined as the solution to y). It is well-known that AdS harmonic functions admit a "split" representation as a convolution of bulk-to-boundary propagators (see e.g. [39,41,56]) Noting that we need to take , and carefully fixing an overall normalization factor (for instance by looking at the coincident point limit), 17 one recovers the identity (3.15).
Let us also note that a single bulk-to-bulk propagator (as opposed to the difference) can be written, using (A.1) and (A.2), as The formula (3.15) for the difference of boundary conditions can then be seen to arise from just (twice) the contribution of the pole at ν = i(∆ − d/2) in the spectral integral above.
As an additional remark, note that the split representation (A.3), coupled with our result for the "two-triangle" diagram arising from the difference of boundary conditions, implies that a given exchange diagram with external operator Φ and exchange of a scalar with dimension ∆ can be written as a sum ofD-functions (one for each channel) as in (3.9), with ∆ in theD-function indices replaced by d/2 + iν, and integrated over the spectral parameter ν with measure determined by (A.3).

B Identities for functionsD, H and G
Let us recollect here explicit definitions and relations between the functions commonly appearing in the AdS/CFT literature. We follow the notations of [30,50]. 17 The precise proportionality constant between the harmonic function and the difference of bulk-to-bulk propagators is found to be

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In AdS/CFT calculations, the D-functions are associated to Witten diagrams involving contact interactions [28][29][30]. At the four-point level x; x 4 ) , (B.1) where we defined the "un-normalized" bulk-to-boundary propagators (B. 2) The integral in (B.1) may be evaluated introducing Schwinger parameters, and yields 3) This can be written in terms of the "reduced"D-functions, which are functions of crossratios only, as [30] In the definition above, the powers ∆ i are arbitrary. In the special case Σ = d/2, the samē D-functions arise from the well-known four-point conformal integral as can be seen by comparing the Schwinger parameter integral in the second line of (B.5) to (B.3). TheD-functions can be related to the H function defined in [30,50] as follows: where Σ = 1 2 i ∆ i and the function H is given by:

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The function G may in turn be defined by explicit power series around u = 0, v = 1: For (positive) integer γ we also get log u terms in the H function, arising from the gamma functions and Pochhammer symbols in the formulas above, H(α, β, k, δ; u, v)| log u = (−1) k (k − 1)! Γ(α)Γ(β)Γ(δ − α)Γ(δ − β) Γ(δ) G(α, β, k, δ; u, 1 − v) (B.9) These will be required to reproduce the small-u behavior of the sum over the conformal blocks. The power series part of the H function is also modified for integer γ. We will use following result for γ = 1 [50]: Also note that the G-function obeys a small-u expansion TheD-functions obey the following symmetry relations:

C Some calculational details
In this appendix we collect various odds and ends of the calculations in sections 3 and 4.
C.1 Adding the ΦΦ → ΨΨ channel to (3.31) The result for the change in the four-point function ΦΨΦΨ in the ΦΦ → ΨΨ channel can be written as: where O(x) =φ i φ i is the singlet operator with tree-level dimension ∆ 0 = d − 2, and S 0 = d d x∂ µ φ i ∂ µφi .
We will need to first study the two-point function of Φ = φ i φ i and Φ * =φ iφi to the first order in perturbation theory: where C 2 and C 3 are two-and three-point function coefficients respectively in the unperturbed theory: The values of C 2 and C 3 are obtained by direct Wick contractions in the unperturbed theory: where C φφ is defined as: where the 2 in the denominator follows from our definition of the φ field: φ i = ϕ i,1 + iϕ i,2 where ϕ i,a , i = 1, . . . , N , a = 1, 2 are real scalar fields with canonical normalization. Using the integral

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we find, after plugging d = 4 − and ∆ 0 = 2 − into (D.2) to the leading order in , We may now exploit the fact that we are only interested in the anomalous dimension at the conformal fixed point of the O(2N ) model, λ * = π 2 N +4 [74], such that the full two-point function in the interacting theory has to be power-law: where γ(λ) = ∆ − ∆ 0 = 2γ φ + γ Φ and . . . stands for terms which either do not have logs or are higher order in λ. Then we may identify the logarithmic terms to find γ(λ) to lowest order: 18 Notice that γ φ is of order 2 , as well known, and doesn't contribute at lowest order. Then the final result can be matched to the well-known anomalous dimension of the symmetric traceless tensor of the O(2N ) model [46,74]. A similar calculation for the ΦΦ * operator may be carried out repeating the same steps as above, whereC 2 andC 3 are now defined through: (D.10) The z-dependence of the three-point function is the same as in the previous calculation, and so will be the value of the integral. To extract γ ΦΦ * we also recall the definition of the anomalous dimension for a composite operator: ∆ ΦΦ * = 2∆ 0 + 4γ φ + 2γ Φ + γ ΦΦ * . (D.11) Here again, γ φ will not matter, being of order O( 2 ). Expanding the exact two-point function and calculatingC 2 andC 3 , we get: in agreement with (5.17). 18 A more accurate argument introducing the multiplicative renormalization of Φ would lead to the same result.

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