Loops in Anti-de Sitter Space

We obtain analytic results for the four-point amplitude, at one loop, of an interacting scalar field theory in four-dimensional, Euclidean anti-de Sitter space without exerting any conformal field theory knowledge. For the two-point function, we provide analytic expressions up to two loops. In addition, we argue that the critical exponents of correlation functions near the conformal boundary of anti-de Sitter space provide the necessary data for the renormalization conditions, thus replacing the usual on-shell condition.


I. INTRODUCTION
Over the past 60 years there has been tremendous progress in the calculation of scattering amplitudes in quantum field theory, in particular, concerning higher loop amplitudes in Yang-Mills theory and (super)gravity.
At the same time we have very few analytic results for loop amplitudes in curved space-times.More precisely, while their short distance expansion and in particular, the structure of counterterms they give rise to is rather well known [1,2], we know little about their dependence on coordinates beyond that [3].Even in de Sitter (dS) or Anti-de Sitter (AdS) space, which are maximally symmetric, admitting the same number of isometries as Minkowski space, little is known about such amplitudes, see [4][5][6][7] for recent attempts.The reason for this is that while in Minkowski space the momentum representation leads to a hierarchy of elementary integrals, in dS or AdS this is not the case and the coordinate representation generally leads to integral expressions that are more manageable in (A)dS.Still, except for some special cases, we lack the technical tools for performing the integrations completely.
In this letter we report on some progress considering the simplest interacting renormalizable field theory.Concretely we compute the two-and four point functions for λφ 4 theory [8] to the second order in the coupling λ on the Poincaré patch of Euclidean AdS 4 by evaluating the corresponding one-and two-loop integrals in coordinate representation.Working on AdS we avoid complications that arise from IR effects on dS for instance [3].It turns out that even this simplified setting is beyond reach for external legs at generic points in AdS but for insertions on the conformal boundary we are able to get explicit expressions.
Being able to go beyond the short distance expansion we encounter an interesting complication concerning the renormalization conditions: For distances that are small compared to the curvature scale the problem reduces to that in flat space and the physical masses provide the right boundary conditions for the renormalized propagator, for instance.At scales of the order of the curvature radius, however, there is no meaningful definition of the mass of a scalar field and one needs to identify a reasonable renormalization condition.In the present case we will find that the AdS/CFT correspondence [9][10][11] provides just that.Indeed, the bulk amplitudes on AdS with external legs inserted at the boundary define a crossing symmetric point correlation function of some primary operator of a conformal field theory (CFT) on the conformal boundary by construction, and therefore a consistent CFT.Of course, we do not know what is the microscopic realization of this CFT, neither do we need it.What matters is that a primary operator has a well defined dimension and this is what replaces the physical mass at large distances [12].
Concretely, let us consider a scalar field with classical action [13] on the Poincaré patch of hyperbolic space of radius 1 a with metric There are two admissible boundary conditions for the classical scalar field corresponding to the asymptotic behavior φ(z, x i ) ∼ z ∆ ϕ(x i ) with a 2 ∆(∆ − 3) = m 2 .Here we will focus on the conformally coupled scalar for which m 2 = −2a 2 and therefore ∆ = 1, 2. However, ∆ = 1 displays some pathology at the quantum level and we thus consider only the case ∆ = 2.The scalar propagator is then given by arXiv:1804.01880v1 [hep-th] 5 Apr 2018 where K is the invariant bi-local function with coordinates x µ = (z, x i ) and y µ = w, y i .Taking one point to the conformal boundary, z → 0, then reduces to the usual bulk-to-boundary propagator [11].

A. Two Point Function
To order λ 2 the 2-pt.function contains the following fundamental constituents: where the first diagram is just a mass shift which will, however, play a prominent role in the following.It corresponds to the elementary integral (here )

Tadpole Diagrams
The 1-loop tadpole diagram requires regularization at short distances, K → 1.We choose which cuts out a small -ball around the pole in the propagator and rescales it by 1 1+ .With this the tadpole diagram reduces to a mass counterterm as expected, For the 2-loop tadpole diagram we adopt the same regularization.It is then possible to show that for z = 0 the y-integral is well defined and is independent of x.Consequently, the nested integral (11) factorizes as where I 2 is again the mass shift and where, after setting x = (1, 0), with Q = y 2 + w 2 + 1.The symmetry w → −w permitted to double the integration region.We then use the Schwinger parameters which allow for a simple spacetime integration, so that For small , the solution for the double tadpole diagram is then given by

Sunset Diagram
Finally we consider the sunset diagram Let us first consider which on the boundary reduces to Here, first we used invariance under translations to set x 2 = (0, 0) and then inversion [14], where z = z x 2 +z 2 .In terms of Schwinger parameters we then have where γ > 0 is an auxiliary parameter.We then perform the spacetime integration before integrating over the Schwinger parameters.Finally differentiating twice w.r.t.γ and subsequently setting γ to 1 we end up with In order to recover the covariant form of (15) we note that undoing the inversion and translation corresponds to 2z → Kx,x2 .With this J 2 takes the form The full sunset diagram can now be obtained by attaching one more leg to J 2 , (cf.13).At the boundary this leads to where I 2 is the mass shift (7) taken at the boundary.It is not hard to see that this result generalizes to generic points in the bulk of AdS by replacing I 2 by I 2 .
In conclusion, all diagrams that contribute to the 2-pt.function up to the second order in the coupling constant reduce to the mass shift diagram which relates the Lagrangian mass to the conformal dimension of a primary operator, O, when evaluated at the boundary.This suggests to replace the physical mass, which is not well defined in AdS, by the conformal dimension of O as the renormalization condition.Our renormalization condition sets ∆ to 2 at all orders in the perturbation.In what follows, we show that this choice is fully consistent with the 4-pt.function computations.

B. Four Point Function
To second order in λ the one particle irreducible diagrams contributing to the 4-pt.function are The tree level contribution of the quartic vertex to the 4-pt.function, given by has already been calculated for external legs inserted on the boundary.Here we just quote the result [15] where we introduced r ij = |x i − x j | and the conformal cross ratios of the coordinates on the boundary η = r14r23 r12r34 , ζ = r14r23 r13r24 .

Loop Diagram
Here we calculate the 1-loop correction given by the double integral considering again first the simpler integral which, by sending z 3 and z 4 to the boundary, takes the form J 4 ∼ J 4 = a 8 (z3z4) 2 (4π 2 ) 4 ϕ 4 where In fact, since the divergence in ϕ 4 is logarithmic we may set = 0 in the prefactor in what follows.As done above, we translate the points x, x 3 , x 4 , by (0, −x i 4 ), which gives x 4 = (0, 0).Then we use the inversion to simplify the above expression where we denote the inverted points by double primes.One more substitution (w , y i ) = (w, y i + x i ) eventually yields where With the help of the Schwinger parameters together with the auxiliary parameter γ to get rid of the w 4 factor, the last integral can be evaluated to give In order to recover the covariant form we first covariantize and thus Finally we attach the remaining two external legs to (21).Sending all external legs to the boundary we have 2 .Again we then 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 3 i ).This gives where is a generating function with auxiliary parameter γ and It can again be evaluated using Schwinger parameters, leading to where . This is the main result of this letter.To continue, L 4 can be evaluated numerically or, alternatively, order by order in an expansion in ζ − 1 and η −1 .Then the expansion coefficients of (24) contain important physical information, which can be extracted by comparing them with the operator product expansion (OPE) in CFT.

C. Comparison to "Experiment"
In flat space, loop corrections to the tree-level amplitudes contain information about the coupling dependence of the masses of resonances for instance.In AdS, where there is no scattering, the role of physical masses is taken by the dimensions of operators of some CFT dual.In our case the CFT contains the scalar operator O dual to φ.As discussed, our renormalization scheme fixes the 2-pt.function to where . To continue we take z i = z ∼ 0 for all external legs.Then, the expansion of the holographic 4-pt.function in the limit η → ∞ and ζ → 1 reads where the factor "3" in the diagrams merely indicates that there are 3 diagrams of this type that contribute to the correlator.Here we furthermore introduced the renormalized coupling constant by a non-minimal subtraction of the form On the other hand, for a CFT in the same limit, inserting the double OPE in the 4-pt.function of identical scalar operators O of conformal dimension ν gives where C 0 , C k are OPE coefficients and γ, κ are anomalous dimensions of O and the leading primary O k arising in the OPE with weight k.The first term is just the schannel disconnected diagram in (26).This constrains ν to be 2, as expected.Second, all remaining terms in (26) go as η −4 and therefore we conclude that k = 4, which implies that O does not enter the OPE directly [16] but merely as a composite field : O 2 :.Furthermore, (26) only contains the conformal invariant η.This constrains γ to zero up to second order in λ R .As discussed above, this is our choice of renormalization condition which replaces the physical mass in flat space.To continue we consider the expansion of the anomalous dimension κ ) and, analogously of the OPE coefficient C 4 .Then, bearing in mind the constraints found above, the expansion of (28) yields The second term is again a part of the disconnected diagrams, but this time corresponding to the u and tchannels in (26).It follows that C 0 = N 2 φ and C φ .Comparing the double OPE (28) with (26) at first order in λ R gives κ whereas at second order in λ R one finds Note that κ (1) agrees at different orders in λ R .This then not only provides a non-trivial consistency test for our 1-loop calculation (24), but also confirms the validity of the AdS/CFT duality beyond tree level in the bulk.
At this point we should stress that scale transformations on the boundary do not affect the cutoff in the bulk since they merely correspond to isometries in AdS.Thus, the fact that the bulk theory has a non-vanishing beta function does not spoil the conformal symmetry on the boundary.

II. CONCLUSIONS
In this letter we computed quantum corrections to the 2-and 4-pt.correlation functions up to the second or-der in the coupling constant for the simplest scalar field theory in AdS 4 .The obtained results for the 2-and 4-pt.functions are mutually consistent.Furthermore, the holographic 4-pt.function can systematically be expanded in the conformal invariants to reveal the OPE structure of the dual CFT, along with the corrections to both the OPE coefficients and conformal dimensions of primary operators.This was carried out at lowest order in the expansion, disclosing a mathematically consistent dual CFT.In particular, the absence of the stress tensor and of any conserved current becomes explicit.
To summarize, it was shown that for 2-and 4-pt.functions, the loop integrals can be evaluated, up to two loops, to the extent that allows for a systematic evalaution of the OPE structure of its dual theory order by order.This should generalize to more complicated theories comprising spinning fields and derivative vertices.