Quantum ϕ4 theory in AdS4 and its CFT dual

We compute the two- and four-point holographic correlation functions up to the second order in the coupling constant for a scalar ϕ4 theory in four-dimensional Euclidean anti-de Sitter space. Analytic expressions for the anomalous dimensions of the leading twist operators are found at one loop, both for Neumann and Dirichlet boundary conditions.


Introduction and Summary
When exploring quantum field theory (QFT) in curved geometries, de Sitter (dS) and antide Sitter (AdS) are the obvious space-times to consider since they are maximally symmetric, having the same number of isometries as Minkowski space-time.While dS is perhaps better motivated in cosmology as an approximation to the early [1], and late time accelerated [2] Universe, AdS has received much attention in mathematical physics due to the relation to conformal field theory (CFT) on its conformal boundary, as exemplified by the AdS/CFT correspondence [3][4][5].On the other hand, progress in QFT on curved space-times has been hampered, in particular, by the absence of a momentum representation for which the Feynman amplitudes can be represented as elementary integrals.The absence of such a representation is particularly limiting at loop level.Indeed, while the short-distance properties of quantum field in curved space-times can be analyzed systematically, little is known about the influence of curvature at distances of the order of the curvature scale.Consequently, to date, explicit tests of the AdS/CFT correspondence have, to a large extent, been limited to classical fields on AdS; that is, CFTs in the leading order in the 1/N expansion with a handful of examples and new techniques having just begun to appear to tackle the loop corrections, see for instance [6][7][8][9][10][11][12][13][14].
Moreover, since the coupling of the CFT stress tensor to the graviton is present almost universally, one is confronted, when considering the bulk theory beyond the classical level, with the quantization of gravity together with its perturbative pathologies in the ultraviolet.
Of course, when embedded in string theory, these singularities should be resolved, but world sheet calculations of string theory in AdS are mostly beyond reach at present [15].
Another class of conjectured dualities, that was supposed to be simpler than the usual string-like dualities, is between CFTs with matter in vector representations of gauge groups and theories with massless higher spin fields in the bulk [16,17].The simplest example of such a duality is given by a free O(N )-vector model, i.e., a bunch of free scalar fields ϕ i (x), i = 1, ..., N with the O(N )-singlet constraint imposed.This duality involves an infinite number of massless higher spin fields in the bulk that are dual to higher spin conserved tensors J s ∼ ϕ∂ s ϕ in the free scalar CFT.Other options include the critical vector model, the free fermion CFT, the Gross-Neveu model [18,19] and, more generally, Chern-Simons matter theories [20].The problem here is twofold.Firstly, higher spin theories reveal some pathological nonlocalities [21][22][23] that prevent them from having a bulk definition that is independent of their CFT duals (but can be defined as anti-holographic duals of the corresponding CFTs).
Secondly, for massless higher spin fields, ultraviolet pathologies of gravity are amplified with increasing spin, see [24].At present, it is unclear how they can be resolved, except for the conformal [25][26][27] and chiral [28] higher spin theories where the nonlocalities are absent and quantum corrections can be shown to vanish [29,30].
One possible way to get around these problems, in the case of AdS, is to use the conformal bootstrap, in particular, crossing symmetry, to determine the coefficients in the operator product expansion (OPE) and the anomalous dimensions of the corresponding operators in CFT, to make predictions for loop-corrected boundary-to-boundary correlation functions of the dual bulk theory in AdS [10][11][12]31].Here, the input is the first-order anomalous dimensions for the "double-trace operators" inferred from the tree-level bulk amplitudes which, using crossing symmetry, lead to an equation for the second-order anomalous dimensions of the latter.This has led to explicit results for a class of dual bulk theories in AdS 3 and AdS 5 [10][11][12].However, this approach does not naturally generalize to AdS 4 , which is the focus of the present work.
In the present paper we propose to compute loop-corrected correlation functions on the Poincaré patch of AdS without using any particular properties of a CFT dual on the conformal boundary [32].More precisely, we consider the bulk theory in a loop expansion in position space.There is a convenient representation for loop diagrams in AdS in terms of Mellin amplitudes [6].A related approach, followed in [7], is to reduce loop diagrams in global coordinates in AdS to a sum over tree-level diagrams using a discrete Mellin space Källén-Lehmann representation with weight function inferred from the OPE in the dual CFT. 1lternatively, one may exploit the fact that the data defining the Mellin representation of the loop diagram is already contained in the tree-level data [10]. 2ere we will not follow this path.Instead, we simply evaluate the loop diagrams in an adapted representation in terms of Schwinger parameters with refinements due originally to Symanzik [33].In order to avoid pathologies associated to spin-2 and above, we will consider a simple interacting scalar bulk theory.Concretely, we consider an interacting bulk scalar field with action on the Poincaré patch of Euclidean AdS 4 .This theory is perturbatively renormalizable and thus we will not have to deal with any of the pathologies mentioned above.In particular, we do not quantize the bulk metric but instead treat it as a background.From the point of view of QFT on curved space-time this is a natural truncation.On the other hand, this may not seem so natural to a reader familiar with the AdS/CFT literature where the graviton appears naturally as the bulk field dual to the stress tensor of the CFT.However, if we do not insist on locality on the CFT side, there are many CFTs that do not possess a local stress tensor.Among them, the critical point of Ising-like models with long distance interactions, see, for example, [34].Such CFTs should also admit an AdS dual description where gravity is frozen to a classical background.
The idea of truncating the bulk theory to an interacting scalar field is not new even in the context of the AdS/CFT correspondence.In particular, this model was considered in [31] as a bulk dual to a CFT with just one low dimensional single-trace operator.The simplest such CFT is the generalized free field [35], which is characterized by the property that the correlation functions of operators factorize similarly to those of the fundamental fields in the Gaußian model.The corresponding bulk dual is just that of a free scalar field φ in AdS [5].If we denote by O ∆ the generalized free field of conformal dimension ∆, then, using the standard AdS/CFT dictionary, its two-point function is given by where φ is the scalar field dual to O ∆ , φ is its restriction to the boundary of AdS, r 12 ≡ |x 1 − x 2 | and N φ is a normalization constant.Similarly the four-point function of the generalized free field will be given by Crossing symmetry of the left hand side will be automatically satisfied and it then has an expansion in conformal blocks of the double-trace operators Next, we consider a deformation of the generalized free field that does not preserve the factorization property.The simplest such renormalizable deformation, already present in Ref. [31], is the φ 4 theory on hyperbolic space.This deformation should correspond to an interacting CFT with a scalar operator O ∆ , m 2 ∝ ∆(∆−d), but without a local stress tensor.
It is clear that any interaction term in an action for the bulk theory will give a crossingsymmetric contribution to the correlation functions on the CFT side by construction.At present, we will take the conformally coupled scalar field in AdS 4 .There are two possible choices of boundary conditions for φ: ∆ = 2 and ∆ = 1, both being within the unitarity window [36].Due to the extremality of the ∆ = 1 case we expect some subtleties at the quantum level.For the same reason we do not include the φ3 interaction.We will discuss this in more detail throughout the text.
The first prediction for the CFT that is computable at tree-level in AdS is the anomalous dimensions and OPE coefficients of double-trace operators appearing in the OPE of O ∆ with itself.This can be extracted from the exchanges and quartic contact interactions, see, for instance, [31,37,38].We then compute the first quantum corrections to the two-and four-point functions, which includes one-and two-loop diagrams in AdS 4 .At a conceptual level, an important implication of this is that the actual loop calculation in the bulk theory is consistent with the duality [32].In addition, this allows us to extract further CFT data.In particular, we can extract higher order corrections to the anomalous dimension of double-trace operators, as well as their OPE coefficients at next-to-leading order in both the deformation parameter λ and the dimension of the double-trace operators.One result, already noted in [31], is that, while the conformal block expansion of the four-point function of the generalized free field involves primary double-trace operators 3 of all even spin and even dimensions, only the OPE coefficients and dimensions of such operators with spin 0 are affected by the interaction at tree-level. 4 At loop level, however, their dimensions are corrected (see also [10] for AdS 3 and AdS 5 ).
One of the main results of this paper is the anomalous dimensions ∆ 0,l of the operators of the leading Regge trajectory, i.e., having the form O ∆ ∂ l O ∆ .These are the lowest-twist double-trace primaries appearing in the OPE of O ∆ with itself.For ∆ = 2 we find for l > 0, (1.3) where γ = −λ R /16π 2 and λ R is the renormalized coupling.For ∆ = 1 we have, in turn, for l > 0, (1.4) where ψ (1) (l + 1) is the trigamma function.Anomalous dimensions for higher twist operators are also computed but they do not seem to have such a simple l-dependence.
Our results have a direct bearing on higher spin theories in AdS 4 as well.Indeed, all these theories contain a scalar field corresponding to ∆ = 1, 2 and need to have a vanishing φ 3 bulk coupling. 5The quartic vertices begin with φ 4 and contain infinitely many [21].Therefore, our results present a meaningful contribution of the φ 4 interaction to the anomalous dimensions and OPE coefficients of higher spin theories and show how to deal with ultraviolet and infrared divergences in the bulk.
An important question that requires clarification at loop level in AdS concerns the dependence on the renormalization scheme.In this paper we use an ultraviolet cut-off regularization, which manifestly preserves covariance, followed by a nonminimal subtraction.For ∆ = 1, there are additional infrared divergences.A convenient covariant regularization of the latter is provided by continuation in ∆.Another issue related to this is the lack of a simple quantum AdS experiment that determines the renormalization conditions in terms of measurable quantities such as the mass of particles, for instance.In the present context we replace the latter by the dimensions of the operators of the dual CFT which seems to be an appropriate replacement in the context of AdS/CFT [32].
In section 2 we specify our conventions and review the construction of the scalar propagator on AdS.In addition, we list the various bulk correlation functions that will be calculated later on.
In section 3 we derive the one-and two-loop corrections to the bulk-to-bulk two-point function in position space using a Schwinger parameterization.Here we also specify the ultraviolet regularization employed in this paper.For ∆ = 1, we will encounter in addition infrared divergencies whose regularization is also discussed there.
Section 4 contains the computation of the four-point function.The tree-level contribution is well known (e.g., [31,38]), so that we will just recall the result.The one-loop contribution requires a ingenious use of Schwinger parameters.The ultraviolet divergences can be absorbed in the renormalized φ 4 coupling as expected.For ∆ = 1, the integral is superficially infrared divergent as well, but these divergences cancel out.Nevertheless, the ∆ = 1 calculation differs from that for ∆ = 2 by an extra contribution, which is computationally tedious but manageable as a short-distance expansion on the boundary.
In section 5 we then compare the short-distance expansion of the bulk four-point func-tion with the conformal block expansion in conformal field theory.At zeroth order in the bulk coupling it is possible to read off the spectrum of double-trace operators.At order λ one determines the anomalous dimensions which vanish for all but the spin-0 double-trace operators at that order.At order λ 2 things become more interesting.Still, for the leading Regge trajectory we are able to derive a closed formula for the anomalous dimensions.
Section 6 contains the conclusions.An extensive list of anomalous dimensions and OPE coefficients for various spins and twists are referred to the appendix.

Preliminaries
We briefly review the kinematical ingredients that are used throughout the paper.The (d + 1)-dimensional Euclidean anti-de Sitter space, H d+1 , can be embedded into a (d + 2)dimensional flat Minkowski ambient space M d+2 .The space H d+1 is then one of the sheets of the two-sheeted hyperboloid where A = 0, ..., d + 1.Given two points X and Y on the hyperboloid, there is a simple relation between the geodesic distance ρ and the scalar product of X with Y A useful parameterization which, however, covers only half of the space, is given by the Poincaré coordinates ), where z > 0 and i = 1, ..., d.In these coordinates the metric makes the conformal flatness explicit ds 2 = 1 a 2 z 2 (dz 2 + dx i 2 ). (2.4) We then introduce a dimensionless O(d + 1, 1)-invariant quantity, related to the geodesic distance (2.2), K := − 1 a 2 X•Y which, when expressed in Poincaré coordinates, reads where x µ ≡ x = (z, x i ) and y µ ≡ y = (w, y i ).The points at z = 0 are said to be "at the boundary" of anti-de Sitter space, and for further reference we note that in the limit where z approaches 0 we have where is the usual bulk-to-boundary propagator [5].
The flat space limit is obtained by letting a → 0. By introducing spherical coordinates with the radial coordinate defined by r = X 2 + X 2 d+1 , K can be shown to have expansion In order to evaluate the Feynman bulk-to-bulk diagrams we need the bulk-to-bulk propagator Λ(x, y; m) for the scalar field.By definition, Λ(x, y; m) satisfies where m is the mass of the scalar field.Making use of the fact that Λ is a function of the geodesic distance (or equivalently K), we find that (2.10) The properly normalized solution to (2.9) is well-known (e.g., [40]) and for d = 3 it reads where ∆ corresponds to the conformal dimension of the dual operator: (2.12) Bearing in mind the above relation, let us henceforth express the propagator by Λ(K; ∆) as a function of K and ∆.In the flat space limit, Λ(K; ∆) reduces (up to a sign) to the well-known scalar propagator in R 4 (2.13) Our model is a conformally coupled scalar field with a quartic self-interaction propagating on a static H 4 background, described by the action where λ > 0 is a dimensionless coupling constant and R is the Ricci scalar.Therefore, in what follows we consider m 2 = −2a 2 , which corresponds to the dimensions ∆ = 1, 2. Then, the propagator simplifies to (2.15) Before closing this section let us express schematically the expansions for the two-and the four-point function respectively, ) where we adopted the following Feynman rules: • each line x y corresponds to the free two-point function Λ(x, y; ∆), • each four-vertex stands for an integral d 4 x |g| over the vertex point x.
One of the purposes of this work is to calculate all the above diagrams.In the following two sections (3 and 4), we will focus on the one-particle irreducible diagrams, since all above diagrams are simple products and/or permutations of the latter.At the beginning of each of these sections, there will be a figure displaying all the computed diagrams in the given section.

Two-Point Function
In this section we compute the one-particle irreducible diagrams that contribute to the twopoint function (2.16), or explicitly As a warm-up we compute the mass shift diagram I 2 , and proceed afterwards in the calculation of the tadpole diagram H 2 and the double tadpole diagram L 2 .Eventually, we discuss the technically more challenging sunset diagram K 2 .For ∆ = 1 we will encounter infrared divergences which are absent for ∆ = 2.

The mass shift diagram
Strictly speaking, this diagram is not needed in our analysis.Nevertheless, it will appear as a counterterm in other diagrams.It is therefore convenient to have its form in order to identify such terms.The mass shift diagram I 2 depicted in (3.1) corresponds to the following integral Using the global AdS symmetries we can move the two points (z 2 , 0) (the covariant form will be restored later).By further denoting x 2 ≡ x i 2 , the integral is then where we have used that |g(x)| = 1 a 4 z 4 .
A covariant regularization of this divergence can be obtained by continuation in ∆, for instance.However, for noninteger values of ∆ the scalar propagator on AdS is complicated.
Still we can proceed, using the fact that for any covariant infrared regularization, I 2 has to obey the inhomogenenous differential equation For ∆ = 1, the most general covariant solution is where the homogeneous part with constants c 1 , c 2 corresponds to free ∆ = 1, 2 propagators.Now consider the boundary limit: The divergent term, whenever an AdS invariant IR regulator is available, should enter in The term proportional to c 2 produces a fall-off behaviour corresponding to the ∆ = 2 boundary condition.For ∆ = 1, we set c 2 = 0.The divergent part should thus be parameterized by c 1 .Therefore, where σ is the IR-regulator and where we made a suitable field redefinition in order to write the boundary limit as For ∆ = 2 there are no IR issues and the evaluation of (3.3) can be carried out straightforwardly.Performing the z-integral and then integrating over x using threedimensional spherical coordinates yields corresponding to the covariant expression (3.10) This solution features two properties.Firstly, it solves as well the differential equation (3.4).
Secondly, its behavior at the boundary exhibits a clear similarity to the regular part of (3.8): In particular, the leading terms of both (3.8), (3.11) are of the order K ∆ .

The tadpole diagram
The tadpole diagram H 2 given in (3.1) has the integral expression The expression Λ(x, x; ∆) is clearly ultraviolet divergent, and needs to be regularized.In principle, we can set it to zero by hand, but we would like to introduce the regulator that we systematically use later on.In position space, UV divergences result in the limit of colliding points, where K → 1.The following "rescaling" is AdS-invariant and resolves the short distance singularity of 1/(1 − K)-like expressions With the help of (2.8), we find that in the flat space limit the regularization takes the form From this it becomes clear that the above regularization carves out a small -ball around the point and then rescales it by 1/(1 + ).This regularization procedure will be used systematically on every UV-divergent integral encountered in this work.More precisely, for propagators representing internal lines we rescale each K as in (3.13), i.e., Λ(x, y; while for propagators representing external legs, only those K's appearing in the numerator are rescaled.Here K xy stands for K as a function of the points x and y.
Returning to the tadpole diagram, the propagator at coincident points is given by Therefore, the tadpole diagram reduces to the mass-shift diagram (3.2) times a divergent prefactor (3.17) After sending x 1 and x 2 to the boundary, (3.17) simplifies to where I 2 is given in (3.8), (3.11).

The double tadpole diagram
The double tadpole diagram L 2 in (3.1) corresponds to the following integral which, containing two loops, requires again a regularization.Adopting the regularization described in section 3.2, L 2 takes the form Let us first consider the integral over y.By the substitution (w, y i ) → (w, y i + x i ), it displays manifest independence on the "nonradial" coordinates x i .Then, a simple rescaling argument can be used to show that the integral also does not depend on the radial coordinate z.Indeed, any rescaling z → θz, θ > 0 can be undone by a substitution of the form y i → θy i , w → θw.
In particular, we can set z to any value z 0 > 0.6 Thus, the nested integral (3.19) factorizes as where I 2 is the mass shift computed in section 3.1 and where In the above formula we define Q = y 2 + w 2 + z 2 0 and use the w → −w symmetry of the integral.For ∆ = 1 this integral exhibits again an IR divergence.We regulate this integral in terms of a parameter σ, Implemented this way, not only the IR-regulator does not affect the ∆ = 2 case but additionally it does not break the rescaling symmetry of z 0 discussed above.
or equivalently In the last step we use the w → −w symmetry and the identity The easiest way to deal with integrals of this form is to implement the, in quantum field theory ubiquitous, Schwinger parameters.Introducing one Schwinger parameter for any of the three factors in the denominator, the integral (3.23) reads Now, the spatial integral is a straightforward Gaussian integral resulting in where we substituted t 3 → (1 + )t 3 .Let us introduce the following coordinates: t i = ss i , which simply correspond to a rescaling of our original coordinates t i by a factor s.
Accordingly, one can rewrite the measure as assuming that the condition s = i t i is satisfied.Hence, (3.25) becomes where we already integrated over s 3 .Integrating over s yields which further leads in the limit σ → 0 to and Eventually, the full double tadpole diagram is given by Note that here, in contrast to the mass shift diagram in section 3.1, the infrared regulator σ appears in the final result.However, it does not spoil covariance.This is because it merely enters (3.31) as a multiplicative factor.∆ = 2.By setting ∆ = 2 and rescaling z 0 to 1, (3.21) becomes Again, it is favourable to introduce Schwinger parameters.This results in which allows for a simple spatial integration, yielding After the substitution t i → ss i , the integral becomes Then, by removing the Schwinger parameter s and subsequently integrating over s 2 , one gets (3.36) Eventually, the evaluation of the last integral yields For small , the full solution for the double tadpole diagram reads We close this subsection with a comment on the renormalization scale.When analyzing QFT in AdS typically one encounters a separation of scales into a UV scale which is related to the scale of local physics and an IR scale given by the AdS radius.In the present context, however, we will be interested in boundary-to-boundary correlation functions for which the AdS radius is the only relevant scale and the UV scale is absent.This is also implicit in the choice of the dimensionless regulator in (3.13), which is related to a dimensionful cut-off Λ through Λ = /a.

The sunset diagram
The sunset diagram K 2 in (3.1) instructs us to compute Let us split it into two parts, the first one being As we are eventually interested in the anomalous dimensions of the operators on the boundary we can already take the boundary limit for x 2 .Then, (3.40) takes the following form In the last step we also introduced the UV regulator, cf.(3.15).Using translation symmetry, we can shift x and x 2 by (0, −x i 2 ), which leads to where x 2 = (0, 0).Next, as was done in Ref. [41], we use inversion symmetry7 to simplify the above expression.In particular, we invert every point by itself, which therefore results in sending x 2 to infinity.For the propagators, this gives where we denoted the inverted points by double primes.Note that the measure of the integral does not change under the inversion.A subsequent variable substitution (w , y i ) = (w, y i + x i ) eventually gives where Q = y 2 +z 2 +w 2 and we used the fact that the integrand is symmetric under w → −w.
∆ = 1.For ∆ = 1, (3.44) becomes A decomposition in partial fractions yields Using the w → −w symmetry, one can write J 2 = J a 2 + J b 2 , where and The computation of J a 2 is rather simple.Indeed, the introduction of one Schwinger parameter yields allowing for the integration over space,

.50)
For small values of , this simplifies to For the calculation of J b 2 , the introduction of two Schwinger parameters is more convenient, Integration over space is now viable and results in (3.53) Notice that the integral does not change if we swap t 1 and t 2 .Therefore one can replace the t 1 in front of the exponential with (t 1 + t 2 )/2.After introducing new coordinates t i = ss i as done above, the integration over s 2 leads to Removing the Schwinger parameter s allows one to integrate over s 1 .Therefore, the final result for J b 2 is given by The integral (3.44) is instead given by As usual, let us introduce the Schwinger parameters where in the last step we introduced an auxiliary parameter γ > 0 in order to get rid of the factor w 4 in the numerator.The spatial integration yields Let us now apply the substitution t i = ss i .After integrating over s 2 , we get Integrating over the Schwinger parameter s gives (3.59) Differentiating twice by γ, setting γ = 1, and subsequently integrating over s 1 , yields which is manifestly covariant.Therefore, it follows that one can write J 2 as Furthermore, due to the covariant form of J 2 in (3.40), it has to correspond to the above result modulo some function of K xx 2 .Considering also the symmetry between x 1 and x 2 in (3.39), one concludes Attaching the missing leg.The full sunset diagram can be obtained by attaching one more leg to J 2 we just extracted the most singular part of, i.e., Then the final result can again be expressed in terms of the mass shift 4 Four-Point Function In this section we compute the diagrams that contribute to the four-point function.Up to the second order in the coupling constant λ, the one-particle irreducible diagrams are The contact cross diagram I 4 is well-known in the literature, see, for instance, [42], and we just quote the result.Afterwards, we compute the one loop diagram K 4 .

The cross diagram
The cross diagram leads to With all external legs on the boundary the integral reduces to with As opposed to the rest of this paper, note the extra factor between I 4 and I 4 which we introduced in order to conform with the literature.The result for I 4 is [42] with the conformal invariants defined as For our purposes a more convenient form for I 4 is given by [43] where we introduce the new invariants

The one loop diagram
The one loop diagram K 4 depicted in (4.1) is given by the double integral which, by sending z 3 and z 4 to the boundary, takes the form As before, the UV divergences are regularized with the help of : Furthermore, since the divergence is logarithmic in , we can safely ignore the prefactor (1 + ) 4−6∆ in what follows.Translating the points x, x 3 , x 4 , by (0, −x i 4 ) yields where x 4 = (0, 0).As detailed in Ref. [41], we use the inversion trick to simplify the above expression.For the K's this gives where the inverted points are denoted by double primes.One more substitution (w , y i ) = (w, y i + x i ) eventually yields Again, we used the symmetry of the integrand under w → −w.∆ = 1.For this value, (4.13) takes the form The Q in the numerator can be written as which allows for a splitting of the integral in a divergent part and a regular part: J 4 = J d 4 +J r 4 .Explicitly, one gets and where for the latter integral, being regular, we already set → 0. Additionally, the regular integral can be further simplified to by using and the symmetry w → −w of the integrand of (4.17).The computation of J d 4 is rather simple.Introducing the Schwinger parameters gives Rescaling the t 1 parameter by the factor 1 + and integrating over space yields After the change of variables t i = ss i , one then finds 3 ( + 1) 2 + z 2 , which for small values of reduces to The same steps done above, applied on J r 4 , lead to Then, the substitution yields a simpler form of the integral where we defined α = |x 3 |/z .This integral admits a closed-form solution in terms of the dilogarithm Li 2 : However, in order to compute the complete four-point function (4.9), the integral form (4.24) will be better suited.
For ∆ = 2, (4.13) becomes instead As usual, let us introduce the Schwinger parameters: (4.27) Integrating over the nonradial coordinates, and subsequently substituting t 1 = ss 1 , t 2 = ss 2 , t 3 = (1 + )ss 3 , yields Additionally, we also introduced above an auxiliary parameter γ > 0 to get rid of the w 4 factor.Then, further evaluation of the integral leads to Differentiating twice by γ, setting γ = 1, and integrating over s 1 , yields Eventually, after integrating over s 3 and taking only the leading orders in , one gets Recovering the full covariance.To get back the explicit covariant form, let us first note which is manifestly covariant.Therefore, it follows that where Attaching the missing legs.In order to obtain the the complete one loop diagram K 4 given in (4.9), we still need to perform the remaining integral Let us send z 1 , z 2 to the boundary, reducing K 4 to In analogy to what was done above, let us translate x k , k = 1, . . ., 4 by (0, −x i 4 ) (denoted by primes), invert all points (denoted by double primes), and then make the substitution (z , x i ) = (z, x i + x i 3 ).Then, the above expression further simplifies to with In order to solve K 4 , let us first introduce a generating function with parameters γ, t > 0: comprising all different cases in (4.35).Indeed, (4.35) can be written as with Introducing Schwinger parameters, integrating over spatial coordinates and making the usual , where we reintroduced the conformal invariants already defined in (4.6).Making another substitution s 1 → ss 1 , s 2 → ss 2 , s 3 → s and integrating over s further simplifies the integral to The change of variables s 1 = t 2 sr, s 2 = t 2 s(1 − r) compactifies one integration region, which leads to Note that, as one might expect, (4.40) is invariant under the exchanges x 1 ↔ x 2 and x 3 ↔ x 4 .
For example, , and then invariance of the above formula follows after a change of variables in r.
Term-by-term computation.Setting γ = 0, t = 1 in (4.39) should lead to the four-point function at tree level, which was computed earlier.Indeed, integrating over s 1 yields corresponding to I 4 given in (4.5) up to a Pfaff transformation of the hypergeometric function.
This result, together with (4.37), confirms the expectation that the UV divergence can be completely absorbed in the coupling constant λ.The first term in (4.35) is given by where ψ(x) denotes the digamma function and where In the ∆ = 1 case, (4.35) contains also the term where By means of the relation which holds for all s ≥ 0, one can rewrite L 4 (η, ζ) as Putting everything together, we find that (cf.(4.37)) to the one-loop four-point function in AdS for ∆ = 1 as well as ∆ = 2.The latter was briefly reported in [32].The t-channel can be simply recovered by an exchange x 1 ↔ x 4 , which turns out to be equivalent to η ↔ ζ.Analogously, the u-channel corresponds to the exchange x 2 ↔ x 4 , which, in terms of η and ζ, translates to η → 1 η , ζ → ζ η .In the next sections we will relate these results to the conformal block expansion which, in turn, defines the dual conformal field theory.This will be done with the help of a short-distance expansion of (4.48).

Conformal Blocks and Anomalous Dimensions
In this section we would like to compare the results of the explicit AdS computations with the general expectations arising from the operator product expansion.First of all, let use summarize the final result of the bulk computation.The Witten two-point function is given by 8 where . To continue we take z i ≡ z for all external legs.Then, for the values ∆ = 1, 2, the Witten four-point function reads (cf.(2.17)) 8 The remaining diagrams including tadpoles and sunset diagrams are proportional to the mass shift and thus merely contribute to the relation between renormalized and bare mass of the bulk scalar, as explained in section 3.4. where and for ∆ = 2. (5.4) The above sum runs over the three cyclic permutations π of the variables x, y, z, namely over the s-, u-and t-channel.The appearing quantities are defined as The logarithmically divergent terms in Eq. ( 5.2) can be absorbed in the coupling constant.Indeed, the renormalized coupling constant obtained through a nonminimal subtraction is given by (5.8) Varying the coupling constant with respect to the square root of leads to the beta function known from standard QFT literature.In what follows, we will use the renormalized coupling constant λ R .In addition, we will fine-tune the renormalized mass such that the renormalized bulk theory is conformally coupled to geometry.Since the mass has a nonvanishing bulk beta function this is not an bulk RG-invariant statement.One might thus worry that such a theory cannot be dual to a conformal theory on the boundary.This apparent contradiction is, however, resolved by noting that three-dimensional scale transformations on the boundary correspond to an isometry in the four-dimensional bulk.

Operator product expansion
Our final goal is to compare our results on the AdS side, that is, correlations functions evaluated on the boundary, with the double OPE of the full four-point function on the boundary itself.In order to do so, we have to consider the separate limits v, Y → 0 of the whole holographic four-point function (5.2).For the cross diagram I 0 , the required expansion is already given to all orders in Eq. ( 5.3).On the other hand, for K 0 given in (5.4), the computation is more elaborate.The adopted procedure is described in detail in Appendix A.
The bulk scalar is dual to a real scalar operator O ∆ of weight ∆.In the leading approx- where A 1/2 n,l are the OPE coefficients.The cubic vertex is absent in our model, and for this reason O ∆ itself does not show up in the OPE.In the leading approximation, the four-point function is given by the disconnected contributions, coming from the product of two-point functions where we drop N 2 φ from (5.2), being an overall factor.On the other hand, the four-point function has a conformal block expansion9 where G 0,0 is the contribution of the unit operator and the squares of the OPE coefficients are given in the appendix B for any conformal weight ∆ and any dimension d, see also Ref. [45].Below we consider separately the two cases of interest for us.
The connected bulk diagrams result in corrections to the OPE data: the OPE coefficients and anomalous dimensions, which we would like to extract.It is useful to consider the squares of the OPE coefficients A n,l as functions of spin and conformal dimension: n,l +... = A n,l + γ n,l A n,l + γ n,l A where it is assumed that γ n,l are just numbers.Therefore, the conformal block expansion, up to second order in the coupling constant, reads where G n,l and G n,l are the derivatives with respect to the conformal dimension evaluated at the free-field value of the double-trace operator's conformal dimension, i.e., at 2∆ + 2n + l.
In what follows we perform the conformal block expansion of the bulk results and extract the OPE data.We first discuss the ∆ = 2 case and only afterwards the ∆ = 1 case, since the former leads to simpler results.∆ = 2.The result of the bulk computation is obtained from (5.2) by setting ∆ = 2.
The zeroth-order OPE coefficients correspond to the disconnected part and follow from the general result discussed above The first order anomalous dimensions are easy to extract from I 0 in (5.2): n,l>0 = 0 . (5.14) Only the scalar operators :O ∆ n O ∆ : receive anomalous dimensions and, for the simplest quartic interaction as considered here, the anomalous dimension does not depend on n, see also [31].It is known that such an interaction does not induce anomalous dimensions for the operators with l > 0. The OPE coefficients are not so illuminating.Furthermore, for operators with spin, they can be determined only at the second order since γ n,l>0 = 0, as is clear from (5.13).
At second order the first few anomalous dimensions read (see Appendix B for a more detailed table) 2,2 = − 269 2520 γ 2 .
Note that the loop correction results in nonvanishing anomalous dimensions for spinning operators as well.Indeed, since there is no operator which saturates the unitarity bound, in our model no local stress tensor and not even a conserved current appears.Therefore, none of the operators is expected to be protected.
While most of the anomalous dimensions are quite complicated, by comparison with the expansion of (5.2) we found a simple formula for the leading-twist operators, i.e. (5.15) The general pattern is that all operators with nonzero spin have negative anomalous dimensions, which should correspond to binding energies in the AdS dual picture.Only the scalar operators have positive anomalous dimensions, which is, however, a second-order effect as compared to γ (1) n,l=0 .From (5.15) we can easily work out the conformal spin expansion of the anomalous dimensions of the operators belonging to the leading trajectory and see the latter to be consistent with the general expectations [46][47][48] , J 2 = (l + τ /2)(l + τ /2 − 1) = (l + 2)(l + 1), (5.16)where the twist τ corresponds to 4.
The first-order anomalous dimensions are found to be γ n=0,l=0 = 2γ , γ n>0,l=0 = γ , γ n,l>0 = 0 . (5.17) We observe a similar pattern as for ∆ = 2, except that the anomalous dimension of the very first operator in the OPE, :O 2 ∆ :, jumps from γ to 2γ.At second order our results are more limited as compared to the ∆ = 2 case, the reason being that we did not find an efficient expansion for the integrals L 0 [x, y, z] in (5.6)where ψ (1) is the digamma function, which can be rewritten also as where H (2) l = l k=1 k −2 are the generalized harmonic numbers.It is actually in this latter form that the anomalous dimensions emerge from the OPE expansion of the bulk integrals.
The last term of (5.18) results from all channels of L 0 [x, y, z] and the s-channel of L 0 [z, y, z].
In the large spin limit the anomalous dimension behaves as  (5.20) It can also be seen that the anomalous dimension admits an expansion in terms of the conformal spin.We expect to find a series of the form [46][47][48] γ 0,l = γ 2 k=1 Q k J 2k , J 2 = (l + τ /2)(l + τ /2 − 1) = l(l + 1), (5.21)where the twist τ is 2 and Q k are coefficients to be determined.The last term in (5.18) contributes with −2 to Q 1 .It is interesting that the first term can also be expanded and the coefficients are related to the Euler-Ramanujan's harmonic number expansion into negative powers of the triangular numbers where B 2j (x) are Bernoulli polynomials.
The anomalous dimensions of the operators belonging to the subleading Regge trajectories do not have any π 2 (or polygamma) contributions and are listed in the Appendix B. All of them are negative for l > 0 and positive for l = 0.

Conclusions
In this paper we gave an analytic derivation of the two-loop correction to bulk-and boundary two-point functions for a conformally coupled interacting scalar field in Euclidean AdS, as well as the one-loop correction for the four-point boundary-to-boundary correlation function, by generalizing the usual flat space Feynman perturbation theory to AdS.The final result can be reduced to a single integral expression which is not given by elementary functions.The remaining integral can either be evaluated numerically or, more importantly, be evaluated analytically in a short-distance expansion on the boundary.We have then shown that the corresponding expansion coefficients fix uniquely the data for the conformal block expansion of the dual conformal field theory on the boundary of AdS and no contradiction arises despite subtleties with UV and (sometimes) IR divergencies.The structure of the dual CFT turns out to be that of a deformed generalized free field of dimension ∆ = 1 and ∆ = 2.The OPE of the CFT contains an infinite number of further primary double-trace operators which have anomalous dimensions and anomalous OPE coefficients that we are able to compute from our boundary correlation functions.This is the AdS equivalent of determining the masses and branching ratios in flat space-time.In order for the interpretation of our result to work out correctly in terms of a dual CFT our loop corrected boundary correlations function have to pass some nontrivial consistency tests.For example, the first order anomalous dimension enters not just at tree-level, but also in the bulk four-point function at one loop multiplying log(v) 2 .Similarly, conformal spin expansion [46][47][48] implies a certain asymptotic fall-off behavior of the anomalous dimensions for large spin.All of these conditions are fulfilled by our bulk correlators.In addition, our bulk calculation gives manifestly finite results for all anomalous dimensions in terms of the renormalized bulk coupling, something that is more difficult to achieve in an approach that reconstructs the bulk correlators from the boundary CFT (e.g., [10][11][12]).

. 60 )
Recovering the full covariance.In order to get back the explicit covariant form, let us first note that undoing the inversion and restoring the translation invariance instructs us to replace 2z → Kxx 2 , (3.61)
imation, O ∆ is a generalized free field and the OPE of O ∆ with itself contains double-trace operators O n,l of all even spins l with conformal dimension 2∆ + 2n + l.Schematically, these operators are of the form O n,l =:O ∆ n ∂ l O ∆ :.The OPE reads , for :O ∆ ∂ l O ∆ : (being in the first Regge trajectory): at high orders in v and Y .Nevertheless, the anomalous dimension of the operators on the first Regge trajectory can be determined to all orders in the spin γ γ