The Conformal Anomaly in bCFT from Momentum Space Perspective

We study the momentum space representation of energy-momentum tensor two-point functions on a space with a planar boundary in $d=3$. We show that non-conservation of momentum in the direction perpendicular to the boundary allows for new phenomena compared to the boundary-less case. Namely we demonstrate how local contact terms arise when the correlators are expanded in the regime where parallel momentum is small compared to the perpendicular one, which corresponds to the near-boundary limit. By exploring two-derivative counterterms involving components of Riemann tensor we identify a finite, scheme-independent part of the two-point function. We then relate this component to the conformal anomaly $c_\partial$ proportional to the boundary curvature $\hat{R}$. In the formalism of this paper $c_\partial$ arises due to integrating out bulk modes coupled to the curved space, which generate local contributions the effective action at the boundary. To calculate the anomaly in specific (free-field) examples, we combine the method of images with Feynman diagrammatic techniques and propose a general methodology for perturbative computations of this type. The framework is tested by computing $c_\partial$ on the explicit example of free scalar with mixed boundary conditions where we find agreement with the literature.


Introduction
The exploration of conformal/trace/Weyl anomaly in even dimensions has a long history [1]. In the simplest case such anomaly appears when a 2d conformal field theory (CFT) without boundary is put on a curved manifold so that the trace of energy-momentum tensor (EMT) receives an anomalous contribution where c is the central charge of the theory. For a given conformally invariant theory it is possible to compute c unambiguously from the operator product expansion of the EMT. The central charge decreases under RG flow [2] and thus serves as an effective measure of degrees of freedom. Some information about the flow of c can be gained by Fourier transforming the EMT two-point function and looking at its low momentum behaviour Roughly speaking this means that the IR limit of EMT correlators 'knows' about the degrees of freedom integrated out throughout the flow. We will see later on that this concept proves to be relevant in studying boundary anomalies too. The situation becomes drastically different in 3d, where no conformal anomaly appears. 1 Nevertheless, one might ask whether an anomaly similar to (1) arises when we insert 2d boundary (or defect) in a 3d CFT. In this case the symmetries of the theory allow [3] for a conformal anomaly localised at the boundary whereR is the boundary Ricci scalar andK 2 = K 2 ij − 1 2 K 2 is conformally invariant quantity depending on the extrinsic curvature K ij . The quantity b is a B− type conformal anomaly in the classification of [4], whereas c ∂ just like c in (1) is an A−type anomaly corresponding to a topological term. By computing heat kernel coefficients of the Laplacian in the presence of boundaries (cf. [5] for review of heat kernel computations and their relation to the conformal anomaly) it was shown (see for example [6] and [7,8] for more recent discussion) that terms of the type (3) appear already in free-field theories. From the holographic perspective (3) can be related to a logarithmic correction to the entanglement entropy [9]. The anomaly c ∂ was found to depend on boundary conditions and therefore to change under boundary RG flow generated by deforming bCFT through addition of some relevant operator at the boundary (which can be understood as flow of boundary conditions as we will see in the next section). This together with the analogy to 2d central charge (1) has lead to a conjecture of monotonicity of c ∂ [7] under boundary RG flow and its proof [10]. The proof uses the anomaly matching techniques of [11] to find a relation between the change of c ∂ along the flow and the two-point function of the boundary EMT. Recently the relationship between correlators on flat spaces with planar boundaries and boundary central charges has been explored in 4d [12] and 3d [13]. In particular in [13] an explicit formula for b anomaly was given by relating the UV divergence in the two-point function of the displacement operator at a planar boundary to a local term ∂MK 2 ij in the effective action. This implies that the b−anomaly is related to the short-distance behaviour of boundary correlators. In this work we will argue that the c ∂ anomaly appears through the the near-boundary behaviour of bulk EMT correlators. In this sense the approach of the present work is closer in spirit to the papers [14], [15] and also [16], where an 'extra' renormalization appears due to divergences from the near-boundary limit. We will develop an intuitive understanding in momentum space, where the near-boundary limit corresponds to large perpendicular momentum. The presence of (divergent) gravitational terms (17) in the effective action leads to a local contribution to EMT correlators in this limit. The paper is structured as follows. In the reminder of the Introduction we will review some essential properties of boundary CFTs. In Section 2 we will discuss the behaviour of EMT correlators in momentum space and the related local counterterms. In Section 2.3 we will derive the contribution of counterterms (17) to the trace anomaly. In Section 2.4 we will derive a formulas (48), (50) for c ∂ , by projecting on scheme-independent, traceless component of the EMT two-point function. Finally in Section 3 we extend and apply Feynman diagram techniques to compute the anomaly for free scalar with mixed boundary conditions. The Appendices A, B contain details of the diagrammatic computations for a free scalar field theory with mixed boundary conditions.

A brief review of b(oundary)CFT
In this paper we will consider 3d Euclidean CFT on a flat (or slightly curved) space. This theory will be defined by a traceless, conserved energy-momentum tensor (EMT) T µν . We will put this theory on a semi-infinite space x 3 < 0. We will use Cartesian coordinates where the Roman indices i = 1, 2 run over the coordinates parallel to the planar boundary placed at x 3 = 0. We will also use the vector notation for vectors parallel to the boundary. We can define boundary-preserving diffeomorphisms ξ µ as a subset of standard diffeomorphisms satisfying boundary conditions ξ 3 | x 3 =0 = 0 and ∂ 3 ξ i | x 3 =0 = 0. Using this one can see that the full 3d conformal group SO(4, 1) is broken down to a subgroup SO(3, 1) of boundary-preserving conformal transformations. This group will consist of SO(2) rotations, parallel translations, dilations and the two parallel special conformal transformations. Together we have therefore 6 generators as expected for SO (3,1). For theory to posses this symmetry the EMT needs to satisfy conformal boundary conditions [17] which means that there is no transport of energy/momentum along the boundary. Nevertheless, there can still be flow of energy from the boundary corresponding to the perpendicular displacement operator D defined through the boundary condition In case of non-conformal boundary conditions, the symmetry group is broken down further. In this case the bulk EMT still satisfies the conservation equation ∂ µ T µν = 0, but this tensor doesn't lead to conserved charges for parallel rotations and translations, since for generic boundary conditions there will be transport of energy/momentum along the boundary. It can be shown that this problem is fixed by defining The new EMT will now define conserved charges for parallel rotations and translations provided By comparison with (7) we see that the full conformal group SO(3, 1) is broken unless the l.h.s. of (8) vanishes. The operator O ij therefore describes a flow of energy at the boundary and it is not conserved (i.e. ∂ i O ij = 0) in general. The equation (8) also reflects the intuitive observation that RG flow on the boundary (r.h.s. of (8)) implies flow of the boundary conditions for bulk operators (l.h.s. of (8)). Note that we can always add a conserved 2d EMT t ij to O ij without violating the condition (8). This is equivalent to adding decoupled degrees of freedom at the boundary.
The conformal invariance is restored if O ij is conserved and it can be shown that this happens if O ii = 0 (see [18]). Therefore intuitively we expect to have conformal symmetry whenever boundary degrees of freedom decouple into a 2d CFT living at the boundary. In this paper we will consider the extreme case of empty boundary with O ij = 0, where only bulk degrees of freedom propagate.

A momentum space perspective
This theory will have a well defined correlators for x 3 < 0 so we will analytically extend these functions to whole 3d space and define the corresponding Fourier transforms. The following Fourier transform will be of interest to this paper Note that because of the presence of boundary p 3 is not necessarily equal to −p 3 . To define the Fourier transforms we have used dimensional regularization so that the dimension of the parallel subspace is d − 1 = 2 − 2 . The correlators will be defined by coupling the theory to a (bulk) background metric g µν . Had we worked in d = 3 without a boundary, there would be no logarithmic divergences (or poles in dimreg) in the above correlator. One way to understand this is by observing that such divergences would have to correspond to diffeomorphism, scaleinvariant local counterterms and there are no such terms in d = 3. We can however include diffeomorphism-invariant, scale-dependent ones These would correspond to power divergences and are present in the most generic case. 2

Small momentum behaviour of correlators
Let us consider the Fourier transform of the two point function of exactly marginal (scalar) operator O in d = 3. In a theory without a boundary, the behaviour of the correlator is completely fixed by conformal symmetry [19]. I.e.
The above behaviour cannot correspond to any local contact term. Adding a planar boundary at x 3 = 0 breaks the translational symmetry in the perpendicular direction, which makes the momentum space structure much richer. We expect two-point functions to be distributionally supported at p 3 = −p 3 in this case. We will see later in explicit examples how this new dependence arises, but for now we would just like to restrict to some general arguments using dimensional analysis. Throughout this paper, we would like to focus on special kinematics with p 3 > 0, p 3 = 0. Such specific kinematics should be sensitive to boundary effects since it can be thought about as an offset to the p 3 = −p 3 conserved limit.
Physically the relevant amplitude should correspond to a process where the incoming state gets absorbed by the boundary. We will further consider an 'effective IR' limit p 3 |p|. This limit is special, since it should probe the physics of the near boundary region |x| |x 3 |, |x 3 |. Using dimensional analysis (or scale invariance) one can conclude that where the constants C 1,2 depend on the boundary conditions and aren't necessarily equal to each other since the global rotational symmetry SO(3) is now broken down to SO(2). This function is well behaved for small |p| due to p 3 > 0 acting as an IR cutoff. The absence of extra delta function, allows for new local local terms p 2 , p 2 3 . Indeed, one readily verifies that for example which comes from a local two-derivative term at the boundary. Higher order contributions to (12) will be non-local terms like etc. Additionally, if the correlator (12) contains divergences we can have an extra non-local contribution which is related to appearance of B−type conformal anomaly. Before we proceed, let us offer some intuitive understanding of the small momentum expansion (12). The component p 3 plays a role of the mass parameter in the two-point functions. The intuitive picture is that since there are no propagating boundary degrees of freedom 3 the bulk fields should get 'integrated out' for large p 3 leaving the boundary theory empty up to some local terms required by anomaly matching. It is these local terms, which should give us the boundary anomalies. This is analogical to a 2d QFT theory where all degrees of freedom are gapped and IR theory consists of pure contact term correlators [20].
In the next subsection we will describe how such local terms of the form (13) arise in the twopoint function (9).

Boundary counterterms and new divergences
In this subsection we proceed to discuss the contribution of local boundary diffeomorphism invariant counterterms to the correlator (9). Before we start we need to define a consistent variation principle for functional derivatives w.r.t. g µν , which will follow from setting the boundary conditions for the metric. To be consistent with boundary diffeomorphisms (4) we take Another way to understand the above conditions is that we want to extend the metric to the unphysical region x 3 > 0 to apply the method of images. In particular we will want the g ij to be symmetric (parity-even) w.r.t. reflections x 3 → −x 3 . All the parity-odd components have to vanish at the boundary, which is equivalent to (15). The variational principle should reflect these conditions, which can be achieved by method of images. In particular the first condition in (15) corresponds to a Neumann boundary condition on the allowed contact terms of T ij correlators. This means for example that all the contact terms in the point function of T ij should be replaced is the position of image in the unphysical region. Such delta function has the desired property that it is equal to δ (3) (x − y) in the bulk and satisfies the Neumann boundary condition. With this in mind we can proceed with the discussion. The presence of boundary allows for the following two scale-invariant terms where δ(x 3 ) is the normal delta function localizing to the boundary surface. Both of these terms have two derivatives and are invariant with respect to the boundary-preserving (parallel) diffeomorphisms.
In addition, we can write further boundary terms andR is the Ricci scalar of the boundary metricĝ ij . 4 The extrinsic curvature K ij depends only on normal derivatives of g ij and g i3 at the boundary so the first two terms above are not to be included when imposing the boundary conditions (15). The coefficients of these terms are fixed by the short-distance behaviour of correlators of pure boundary operators. For example the first term was shown [13] to be related to the short-distance divergences of the two-point function of the displacement operator (6). More specifically it was shown that D(x)D(0) contains divergences proportional to ∂ 4 δ (d−1) (x). This contact term has vanishing Fourier transform if we take p 3 > 0 and hence doesn't appear in correlators of the type (12) studied in this paper. 5 In what follows we will focus solely on (17). Apart from (17), (18) there are also other admissible boundary counterterms, but these are either total derivatives or not scale-invariant so we will not discuss them here. The counterterms in (17) might also have divergent coefficients in general. These divergences don't come from the short-distance limit as usual but rather from the near-boundary behaviour. For example if the theory has an operator O of dimension d − 2, the correlator of the trace of EMT (which is pure contact term in CFT) includes the following contribution In a CFT without a boundary such terms vanish since O = 0, but in the presence of boundary Hence for bCFT such contact term becomes semi-local and acquires non-trivial Fourier transform. For example a following contribution could arise Similarly the two-point function of EMT usually contains a term proportional to One possible source of contact terms of the type (19) are for example equations of motion. As we will see in the explicit examples (e.g. in Appendix B) new divergences will appear when evaluating loops for Feynman diagrams with non-conserved p 3 . Such divergences should be removed by adding covariant counterterms localised on the boundary [14]. Even if the corresponding correlator doesn't have divergences, the choice of counterterm coefficient represents a contact term ambiguity. Physical quantities shouldn't involve such ambiguity so it important to understand, which correlators are invariant. To define renormalized correlators in dimensional regularization we will apply the quantum action principle [21]. The two-point functions can be defined through second derivative of the partition function where the effective action (generating functional of connected Green's functions) was defined W = − ln D(φ)e −S and the variational rule (16) is assumed. By the quantum action principle this correlator should be finite. More precisely we can rewrite (22) as where the fist term on r.h.s. of (23) can be thought of as the bare part, whereas the second term includes contact terms necessary to renormalize the correlator. 6 Let us now check how (17) contribute to some particular two-point functions of T µν by writing a counterterm action 6 Note that this definition only includes connected Green's functions.
where an arbitrary mass scale µ was included to preserve the scale-invariance. Expanding these terms to quadratic order in the metric and taking the variations we can find their contribution to various correlators of energy-momentum tensor. In particular we have where we omitted the (non-zero) part proportional to p 2 3 , p 2 3 , since it is not relevant for our purposes. 7 In (25) we have emphasized that c 1 gives evanescent contribution near d = 3 and c 2 vanishes identically. Since (17) are the only counterterms yielding p 2 terms, we immediately deduce two things. First, it follows that in d = 3 the two-point functions of purely parallel components of T µν should not include any divergences proportional to p 2 . The second implication is that this component is invariant w.r.t. scheme change induced by a finite shift in c 1 , c 2 in (24). This is reminiscent of what happens with √ gR counterterm in 2d and is related to the topological character of ∂MR . On the other hand the correlator of perpendicular components receives contribution from (24) We see that the p 2 divergent part of the coefficient c 2 is fully determined from the p 2 divergence of T 33 T 33 and the knowledge of c 1 . As we will see in the next section conformal invariance implies that the p 2 divergent part of T 33 T 33 vanishes. 8 Nevertheless, the (finite) p 2 part of T 33 T 33 is still arbitrary and dependent on the choice of c 1 , c 2 . The correlators involving T 3i receive no p 2 contributions from (24) or any other counterterm due to boundary conditions (15) and the related variation rule (16). These correlators should not have any divergences from the near-boundary limit due to the boundary condition (5).

The trace anomaly
We would now like to discuss how (17) contribute to the trace anomaly. In order to that we will extend the standard dimensional regularization arguments [22] to the case of a theory with boundary. In this approach the (bare) effective action is conformally invariant, but contains divergences W = c + . . . . These divergences need to be subtracted using suitable local counterterms proportional to µ −2 , which break the conformal invariance and the trace anomaly arises. In the case of d = 3 − 2 , there is no bulk anomaly so the divergent part of W should be localised on the boundary We expect this term to arise from near boundary behaviour of the bulk effective action integral.
To guess this behaviour we will isolate the part of the action integral coming from an infinitesimal strip of width δ near the boundary. Using covariance w.r.t. boundary diffeomorphisms and scale invariance we expect the relevant contribution to action to be of the form The bulk action is finite (there are no bulk anomalies in odd dimensions) so both c ∂ , b n should be finite and well defined quantities. Using the identity 1 . . we find that indeed (28) reduces to (27) in the d → 3 limit. The divergence in (28) comes from the near boundary limit x 3 → 0, which is consistent with the discussion in the Section 2.2. Had we used the cutoff regularization x 3 < 1 Λ instead of dim-reg, the divergence would reappear as log Λ. From (28) we see that even though the bulk effective action in non-local its contribution to the boundary part (27) is local (polynomial in p 2 ). This doesn't mean that the boundary part of the effective action is local in general. There will be non-local contributions corresponding to the sources of operators living at the boundary. For example if we took O ij = 0 in (7), the effective action would contain non-local terms depending on g ij at the boundary. By taking two metric variations of (27) and Fourier transforming the result we can identify the anomaly contribution to the two-point function of T ij where and P ij = 1 p 2 (p i p j − δ ij p 2 ). The dots stand for the p 2 3 term and finite contributions generated from the regular part of W . The operator T ijkl is transverse, local in p i and evanescent close to d = 3 as expected already from (25). In fact, tracing (29) we get which is analogical to the contact term appearing in the 2d trace anomaly. 10 Even though the correlators of energy-momentum tensor are finite, the divergent effective action is renormalized via defining W f in = W + S ct , where the counterterm action S ct of the form (24) has These counterterms break scale invariance through the arbitrary scale µ and the trace anomaly appears [23] d We should stress that in (33) we only included bulk contribution to the trace anomaly. In general one should also include the terms (18) in W an . In particular the first term is important 9 When performing the metric variation and Fourier transform we have to be careful with factors of 2. First we pick up a factor of 2 from the 'image' delta function (cf. (16)) and we also pick up a factor of 2 from integrating δ(x3) over the unphysical region x3 > 0. 10 In fact by writing the boundary metric in the conformal gaugeĝij = e −2τ δij we find that the finite part of Wan contains the term d 2 xτ τ , which is exactly the anomaly-induced (dilaton) effective action in a 2d CFT.
for the b−anomaly as was discussed in [13]. 11 To understand the anomalies (33) we can use Gauss-Codazzi relations where the dots stand for total derivative terms and a i = (∇ 3 n) i = Γ 3 3i ∼ ∂ i g 33 is the acceleration vector. Note that unlike the extrinsic curvature , the a 2 i terms don't vanish under boundary conditions (15). However, such term is ruled out in a confromal theory by Wess-Zumino consistency conditions 12 [3] and hence we obtain a constraint By combining this condition with (32) and (26) we conclude that T 33 T 33 has no p 2 divergences. In Appendix B, we will find this relation to be satisfied for conformally coupled free scalar with mixed boundary conditions. Finally, neglecting the other terms involving extrinsic curvature and total derivatives we get which shows that the Ricci scalar anomaly is determined directly from the 1 pole of R in the effective action. In the next section we will show how to relate this coefficient to the correlators of bulk EMT.

Searching for a physical quantity
We would now like to look for a finite, scheme-independent quantities. By scheme we mean the choice of finite coefficients of counterterms like (17). From discussion of the previous section (cf. (25)) we see that the purely parallel component T ij (p, p 3 )T kl (−p, p 3 ) receives no p 2 corrections from (17) so we will project on its p 2 part. The resulting quantity still receives correction from the first term in (10). To remove this ambiguity we can choose a configuration of external momenta which explicitly violates momentum conservation with p 3 > 0 and p 3 = 0.
Finally we see that a desired scheme-invariant quantity should be some combination of terms of the type where ∂ ∂p 2 stands for some second order differential operator in p i that projects on the right kinematic structure. Using again the analogy with 2d massive flow we expect that at small momentum p 2 p 2 3 some combination of components (37) should give us the central charge (cf. (2)). 13 The expansion (12) applied to T ij will have the form It would be tempting now to equate A ijkl with T ijkl from (29). Such argument is complicated by the fact that T ij is not conserved in the parallel direction, which means that non-covariant (finite) contact terms also contribute to A ijkl . In fact, the general form of such correction is which follows from the transverse property p k T kl (−p, 0) = 0 valid at p 3 = 0. 14 We verified that (39) appears already in the free scalar example. The anomaly (29) should equal to the trasverse component of A ijkl so the terms contributing to (39) need to be projected away. We can either do this on a case-by-case basis via shifting A ijkl by a contact term proportional to δ ij (p k p l − δ kl p 2 ) or we can work with correlators of the traceless component to which (39) doesn't contribute. We will choose the latter for more generality and other reasons that we would like to present now. It can be shown by using formulas of [24] that for |x − x | |x 3 | the two-point function ofT ij behaves as (1) is a related to a specific kinematic structure defined in [24] and it depends on boundary conditions. For d = 3 the r.h.s. of (41) looks exactly like the 2-point function of conserved EMT in a 2d CFT. 15 Before we proceed, let us discuss some further desirable properties of the operatorT ij . We saw at the end of Section 2.2 that the two-point functions of T µν receive non-trivial contributions from equations of motion in the presence of boundary. One possible source of e.o.m. ambiguity 16 comes from defining dimensions of fundamental fields under conformal (Weyl) transformations [25] T µν → T µν + cδ µν φ(x) δS δφ(x) .
Such ambiguities do not contribute to the combination (40). Furthermore, the leading term in the expansion (12) of the Fourier transform of the correlator (41) at non-coincident points doesn't depend on bulk operator improvement of the form 13 In the configuration space this limit should correspond to large |x − x | |x3| and thus effectively probing large-distance, near-boundary behaviour of the correlator. 14 Note that this identity is only valid on-shell, however in this case the equations of motion only contribute to the p 2 3 component 15 The extra scale factor (x − x ) 2 on the l.h.s. of (41) appears through the Fourier integrals over dx3, dx 3 . 16 Here we assume a symmetric energy-momentum tensor. In general there are also ambiguities related to Lorenz transformations, which induce antisymmetric improvements.
for some local operator O of scaling dimension (d − 2). This can be shown by noting that the relevant contribution of above improvement to T ijTkl will always be of the form p 2 OT ij . From tracelessness ofT ij , dimensional analysis and SO (2)  . It should be noted that as we will see later on in the free scalar example, the ξ-dependence will still come in through the Fourier transform of the contact term of the type (19) obtained by varying (43) w.r.t. metric twice. Alternatively it can be seen as a mean-field contribution to the divergent part of the effective action (27) Close to the boundary O ∼ 1 |x 3 | d−2 , so the above integral is exactly of the form (28) and therefore contributes to c ∂ . We will now proceed to study the low p 2 expansion of the two-point function whereÃ ijkl ,B ijkl are traceless components of the corresponding quantities on (38). From (39) we can deduce where κ is a model-dependent constant. 17 On the other hand from (22) and (27) it is seen that (46) should correspond to the traceless part of (29). Indeed we by projecting By equating (47) with (46) we finally arive at Due to presence of p 3 > 0, which acts as an effective IR cutoff, the small p behaviour of (45) should be under control and as previously seen all correlators of T ij are scheme-independent and finite so κ is a well defined and unambiguous quantity. 18 Let us now discuss how to compute κ directly from T ijTkl . This can be done for example by taking momentum derivatives of a suitable contraction of (46) One slightly confusing point here is that although we have argued that the non-covariant terms don't contribute to the traceless partÃ ijkl , the relation (46) implies its non-transverse character. The reason for this apparent discrepancy is the fact that traceless part of transverse projector is itself not transverse in d > 1. 18 The scale invariance prevents the κ from depending on p3 in strictly d = 3. = 1 8 This result can also be cast into a sum-rule relation in the position space where the operatorT was defined in (40). Let us recap the main points of this rather technical section. The point of above exercise was to identify the local, transverse structure corresponding to the c ∂ anomaly in the T ij T kl correlator (38). In the Section 2.2 it was argued that the p 2 component A ijkl is unambiguous, therefore a suitable candidate. As seen in Section 2.3 the part of the two-point function related to the anomaly (cf. (29)) is transverse so we had to project out non-transverse contributions (39). We argued that the easiest way to achieve it was to look at correlators of the traceless componentT ij . To simplify the computation and comparison with (29) it was convenient to contract with p i in (46), which lead to the final result (48). The result was then recast into a position space relation (50).

Free scalar
In this section we will illustrate the main ideas of this paper on the explicit example of free scalar field with mixed boundary conditions. In the bulk, this theory is described by the following action where we have summed over N non-interacting scalar fields. The conserved bulk EMT for (51) reads For T µν to be traceless we would need to take ξ c = (d−2) 4(d−1) , but we will use generic ξ in the following to achieve more generality. We can express the mixed boundary condition by using projectors Π − and Π + which project onto subspaces satisfying Dirichlet and Neumann conditions respectively. More concretely by expressing φ A as a column vector and projectors as matrices the boundary conditions read [5] Π − φ| x 3 =0 = 0 , The Green's function satisfying above conditions can be found by using the method of images . We will now analytically extend this function to the upper semi-space x 3 , x 3 ≥ 0 and Fourier transform it. Doing this we will obtain a propagator depicted on Figure 1. The first term is the standard propagator 1 p 2 , with momentum being conserved across. The second term is new-it is also proportional to 1 p 2 but now the perpendicular momentum is not being conserved, which is expressed by the dashed line. This extra term can be intuitively understood as an elastic reflection from the planar boundary.

Perturbation theory with boundary
For any given Feynman diagram we can treat the second term on Figure 1 as the usual propagator and the dashed line as an external momentum emission (even if it is attached to virtual propagators) as long as we include the overall delta function where the sum goes over perpendicular components of all external momenta, including the dashed lines. Including bulk vertices is possible provided they are invariant w.r.t. parity-like transformation φ(x, −x 3 ) → χφ(x, x 3 ), where χ was defined under (54). 19 In this case each vertex has to be multiplied by a factor 1 2 to avoid the double counting that arises from including the x 3 > 0 contribution, but otherwise can be treated as usual conserving all components of the incoming momentum. 20 These rules allow us for systematic treatment of perturbative computations in scalar field theory with boundary. The same rules apply to theory with fermions, the only thing that changes is the form of free propagator and χ. Note that the presence of delta function depending on internal momenta (55) allows for some new phenomena. In particular would-be power divergences now become logarithmic and we need to include new counterterms to renormalize those. Such (local) counterterms will exist purely on the boundary and will lead to some additional renormalization. The main idea can be illustrated on the free-scalar example at hand by adding φ 4 interaction (see Figure 2). The total delta function (55) imposes 2l 3 = p 3 +p 3 for the perpendicular compo- 19 As an example one can take single scalar with λφ 4 since under the reflection it changes as λ → χ 4 λ = λ.
More generally λ has to be χ−invariant tensor. 20 We can also include boundary interactions, that only conserve parallel momentum by using that δ(x3) = 1 2π ∞ −∞ dL3e iL 3 x 3 , so that the relevant vertex will be proportional to dL3e iL 3 x 3 with L3 = incoming p i 3 . The remaining integral will be nonzero and equal the 2d massive bubble diagram with a mass m = 1 2 (p 3 +p 3 ) and thus needs to be renormalized by adding a (divergent) 'boundary mass' counterterm 1 d 3 xδ(x 3 )φ 2 (x). Thus we see that bulk interactions induce renormalization of purely boundary terms. The same conclusion was reached in the original paper [14] and more recently renormalization of boundary couplings in d = 4 − 2 was also discussed in [12]. For more detailed analysis including the discussion of renormalizability we refer the reader to [15]. Next we would like to discuss the renormalization of composite operators. To this end we can use the local coupling formalism, where the coupling to a composite operator O gets promoted to a function of space The correlators of O are then obtained by taking functional derivatives of the partition function w.r.t. λ(x). In [26] it was shown that doing this it is necessary to include additional divergent counterterms dependent on λ(x) and its derivatives. These have to be include in order to cancel UV-divergent contact terms in the composite operator correlators. When including the planar boundary we expect some counterterms proportional to δ(x 3 ) to appear. Indeed, for example by considering again O = φ 4 one observes that the diagram on Figure 3 in d = 3 − 2 now includes logarithmic divergence (a pole in ). Once again we need to include, purely boundary term 1 d d xδ(x 3 )λ(x) 2 to cancel this divergence. A detailed local coupling analysis of scalar field theories with boundaries was given in [16].
In the context of this paper we treat the metric g µν as the local coupling to the energymomentum tensor. The relevant counterterms are then the ones invariant under boundary diffeomorphisms (17).

Energy-momentum tensor correlator
The relevant (bare) two-point functions are defined through the variational principle (22) with the rule (16) assumed The first term on r.h.s. of (57) is defined is simply the correlator of (52) at non-coincident points. In momentum space it corresponds to the two diagrams on Figure 4. The other possible diagrams include the usual d = 3 contribution with no dashed outgoing lines and the other 'reflected' diagram with two outgoing dashed lines. The latter two diagrams are proportional to δ(p 3 + p 3 ) and δ(p 3 − p 3 ) respectively and both of these distributions vanish for the momentum configuration of (A.1). The nonvanishing diagrams of Figure 4 correspond to absorption of virtual scalar by the boundary and can be computed via standard methods (we refer the reader to the Appendix A, where some more details of these computations are given). The second piece in (57) is a contact term. In fact, as we saw in Section 2.4, no purely background counterterms contribute to (57) in d = 3 so the only possible contribution comes from varying the second term in (51). This contribution is new compared to the boundary-less case where φ 2 = 0. We have where D ijkl is a differential operator quadratic in derivatives that is obtained from the variation √ gR. This operator is evanescent near d = 3, which will leave a finite imprint when combined with the divergent Fourier transform of φ 2 (x) ∼ Trχ 1 |x 3 | in (58). Notice that this is an example of 'mean field' contribution discussed around (44). We are interested in the traceless operatorT ij defined in (40) Correlators ofT ij are constructed from the traceless projection of (57) so we now have all the pieces needed for the computation of (46). Evaluating the diagrams on Figure 4 and including the Fourier transform of (58) (see Appendix A) we finally arrive at which yields the central charge where we used (48). The trace anomaly follows from (36) and we find agreement with the result of [5] (including the dependence on ξ). For conformal theory we additionally require ξ c = 1 8 and get For single scalar field the trace is Tr(χ) = ±1 with positive sign for Neumann and negative sign for Dirichlet boundary conditions. In this case (62) reduces to the result of [10] obtained by computing the partition function of conformally coupled scalar on a hemisphere. We also computed the anomaly of free fermions by the same method and verified that it vanishes as seen in the literature [9].

Conclusions and outlook
The main goal and result of this paper was to derive the relations (48), (50) between the central charge c ∂ and flat space EMT correlators (46). We saw that the bulk contributes local divergent terms (27) to the generating functional W at the boundary, which are then related to finite contact term of (45) in the small |p| p 3 regime. This situation is similar to integrating out a massive particle in a 2d (cf. (2) with c IR = 0 and c UV = c), which leaves behind a local term in the IR effective action proportional to the central charge (such setup can be realized by coupling the theory to a background dilaton and computing the resulting IR effective action [27]). In our case the the perpendicular momentum p 3 played the role of mass parameter when integrating out the bulk state to obtain c ∂ . Physically the relevant amplitude (38) corresponds to the process where the incoming bulk state is being absorbed by the boundary. As a side result we obtained a criterion (35) guaranteeing no p 2 divergences in the two point function of perpendicular components T 33 (p, p 3 ), T 33 (p, 0) , which is consistent with the literature. The present method only allows us to find the bulk contribution to c ∂ coming from correlators of T µν coupled to bulk metric with boundary conditions (15). It would be interesting to extend the analysis of this paper to include the computation of c ∂ in presence of non-trivial boundary operators (for example in theories that have non-vanishing O ij with bulk-boundary interactions). Since ∆c ∂ under boundary RG flow only depends on the dynamics of boundary operators [10], it would not be unreasonable to expect that the bulk and boundary contributions simply sum. The second main contribution of this paper is development of a general perturbative framework in Section 3.2, where the boundary can be treated as an external state (see for example Figure 4). Momentum space methods have already been applied to boundary field theories in the literature [15,14,12]. Where this work complements the previous analysis is in including also the Fourier transform in the non-conserved direction. We saw that doing this allows one to use standard diagrammatic techniques and therefore can be applied in a broad context. A natural extension of this work would be to check the method in examples where the bulk is weakly interacting and compute the perturbative corrections to c ∂ . one-loop diagram of 'effective mass' m = 1 2 p 3 in d = 2−2 . After the Feynman parametrisation the divergent piece of this integral cancels out and (A.4) becomes 1 4π Tr(χ)m 2 p j p k p l − 1 (d − 1) δ kl p 2 1 0 x(1 − x)dx x(1 − x)p 2 + m 2 . (A.5) Due to the presence of non-zero mass the above integral has regular behaviour at small p 2 and we are able to get the final result ) . (A.6) Extension to ξ = 0 is easy, since it turns out that the contributions proportional to ξ in the diagrams on Figure 4 only contribute at O( ) and thus we can ignore them for the purposes of this paper. The contribution from the second term in (57) is obtained by projecting (58) to its traceless componentD ijkl , which satisfies We can use this to evaluate the Fourier transform of Using that φ 2 = Tr(χ) 4π 1 2|x 3 | and computing the Fourier integrals over parallel coordinates and dx 3 (with p 3 = 0) we reduce the Fourier integrals to one contribution In this appendix we would like to illustrate how divergences arise in the correlator T 33 (p, p 3 )T 33 (−p, 0) arise. The momentum space vertex for T 33 φφ reads V 33 (q 1 , q 2 ) = (q 1 ) 3 (q 2 ) 3 − (q 1 · q 1 ) + 2ξp 2 . (B.1) The relevant amplitude is obtained from one-loop diagram on Figure 3. Just as in Appendix A, the integration over perpendicular loop momentum is removed by δ(2l 3 − p 3 ) and we are left with d (d−1) l loop integral with effective mass m = 1 2 p 3 . The numerator of the integrand will contain four powers of momenta arising from two vertices (B.1) and we can expand it in terms of inverse powers of propagators P 1 = l 2 + m 2 , P 2 = (p − l) 2 + m 2 . By doing this we find two divergent pieces. The first one proportional to p This term has to be cancelled by adding a divergent counterterm (26). Note however, that for conformally coupled scalar with ξ c = 1 8 this contributions vanishes on its own as expected from the condition (35). The cancellation of divergences is consistent with the results of [24], where the boundary limit of T 33 T 33 (α(v) → α(1) in their language) for conformally coupled scalar contains no divergences and also with the explicit heat kernel computation of [6].