Perturbative RG flows in AdS: an étude

Edoardo Lauria1,2, Michael Milam2,3 and Balt C. van Rees2

1 LPENS, Département de physique, École Normale Supérieure - PSL

Centre Automatique et Systèmes (CAS), Mines Paris - PSL

Université PSL, Sorbonne Université, CNRS, Inria, 75005 Paris

2 CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120
Palaiseau, France

3 Institut de Physique Théorique, Université Paris-Saclay, CNRS, CEA, F-9119,
Gif-sur-Yvette, France

<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

We discuss general properties of perturbative RG flows in AdS with a focus on
the treatment of boundary conditions and infrared divergences. In contrast
with flat-space boundary QFT, general covariance in AdS implies the absence of
independent boundary flows. We illustrate how boundary correlation functions
remain conformally covariant even if the bulk QFT has a scale. We apply our
general discussion to the RG flow between consecutive unitary diagonal minimal
models which is triggered by the $\phi_{(1,3)}$ operator. For these theories
we conjecture a flow diagram whose form is significantly simpler than that in
flat-space boundary QFT. In several stand-alone appendices we discuss two-
dimensional BCFTs in general and the minimal model BCFTs in particular. These
include both an extensive review as well as the computation of several new
BCFT correlation functions.

###### Contents

  1. 1 Introduction
  2. 2 General principles
    1. 2.1 From BCFT to AdS
    2. 2.2 Towards the boundary
      1. 2.2.1 Distance versus curvature
  3. 3 Hyperbolic life at one loop
    1. 3.1 Ultraviolet behavior
    2. 3.2 Infrared divergences
      1. 3.2.1 Covariance and comparison to BCFT
      2. 3.2.2 Comparison to holographic renormalization
      3. 3.2.3 Is AdS a box?
    3. 3.3 Anomalous dimensions of boundary operators
      1. 3.3.1 Identical operators
      2. 3.3.2 Example: anomalous dimension of D in $\text{AdS}_{2}$
      3. 3.3.3 Boundary operators mixing at one-loop
  4. 4 Flows between minimal models
    1. 4.1 Infrared divergences
      1. 4.1.1 Large $m$ limit
    2. 4.2 Anomalous dimensions of boundary operators
      1. 4.2.1 Spectrum of boundary Virasoro primaries
    3. 4.3 Bulk one-point function of $\phi_{(1,3)}$
    4. 4.4 Irrelevant deformation
      1. 4.4.1 The perturbation
      2. 4.4.2 Anomalous dimensions
      3. 4.4.3 Large $m$
  5. 5 Outlook
  6. A Two-dimensional BCFTs
    1. A.1 Global conformal invariance
    2. A.2 Local conformal invariance
      1. A.2.1 Conformal Ward identities on the upper half-plane
      2. A.2.2 The method of images
    3. A.3 Some universal correlation functions
      1. A.3.1 Interlude: the stress tensor on the plane
      2. A.3.2 Correlation functions of the stress tensor
      3. A.3.3 Correlation functions of the displacement operator
      4. A.3.4 Correlation functions of the displacement with boundary operators
      5. A.3.5 Correlation functions of the displacement with a bulk operator
      6. A.3.6 Correlation functions with $\text{D}^{2}$
    4. A.4 Correlation functions with $T\bar{T}$
    5. A.5 A correlation function with $\text{D}^{2}$ and $T\bar{T}$
  7. B Review of minimal model BCFTs
    1. B.1 Conformal boundary conditions
    2. B.2 The $\phi_{(1,3)}$ and $\psi_{(1,3)}$ operators
    3. B.3 The bulk two-point function of $\phi_{(1,3)}$
      1. B.3.1 Bulk-boundary crossing
    4. B.4 Correlator between four $\psi_{(1,3)}$
      1. B.4.1 Boundary crossing
    5. B.5 Correlator between two $\psi_{(r,s)}$ and one $\phi_{(1,3)}$
      1. B.5.1 Removing unphysical singularities
    6. B.6 The $\phi_{(3,1)}$ and $\psi_{(3,1)}$ operators
    7. B.7 The bulk two-point function of $\phi_{(3,1)}$
      1. B.7.1 Solution to bulk-boundary crossing
    8. B.8 Correlator between four $\psi_{(3,1)}$
      1. B.8.1 Boundary crossing
    9. B.9 Correlator between two $\psi_{(r,s)}$ and one $\phi_{(3,1)}$
      1. B.9.1 Removing unphysical singularities
  8. C Some integrals on AdSp+1

##  1 Introduction

This work concerns perturbative and covariant renormalization group flows for
QFTs in AdS. Such flows are interesting because the boundary correlation
functions remain conformally invariant along the entire flow which means they
can be studied with conformal bootstrap techniques [1, 2]. Some early works on
QFT in AdS are [3, 4]. More recently interesting results at large $N$ were
obtained in [5, 6], and the structure of certain correlation functions
involving the stress tensor was analyzed in [7]. Of course, in the flat-space
limit there are many connections to the S-matrix bootstrap [8], both for
gapped [9] and gapless [10] theories, but in this paper we will mostly discuss
theories with non-trivial IR fixed points.

The first part of this paper discusses several general aspects of these flows.
Compared to flat space, one novelty of RG flows in AdS is the non-zero
background curvature. The intricacies this introduces in perturbation theory
are however well-known and discussed in many places; see for example [11] for
a classic reference. We therefore focus on the second novelty of RG flows in
AdS: the treatment of boundary conditions and infrared divergences.

Let us start with the UV. A foolproof way to set up a QFT in AdS, at the level
of correlation functions at least, is to start with a CFT on a half-space and
to apply the usual Weyl rescaling rules to the local operators to obtain their
AdS counterparts. This procedure appears to work for any BCFT setup, so any
conformal boundary condition immediately leads to a consistent AdS boundary
condition.

Next we can consider deformations that break conformal invariance. As we
discuss below, one can expect infrared divergences if the boundary spectrum
involves marginal or relevant operators. Infrared divergences in flat-space
scattering amplitudes signal a problem with the observable itself (more
specifically, individual amplitudes are simply zero), but in AdS the infrared
divergences can be mitigated with suitable counterterms which make both bulk
and boundary correlation functions finite.

In this work we will analyze the structure of these divergences at one loop in
some generality. We note in particular that the boundary action is often fully
fixed by requiring boundary conformal invariance. This observation, which we
can summarize by saying that “the boundary follows the bulk”, is in sharp
contrast to the BQFT setup where the boundary can flow independently from the
bulk theory. Below we also explicitly discuss the renormalization of boundary
two-point functions. We show that they remain conformally invariant but that
extra logarithmic divergences appear which give rise to anomalous dimensions
of the boundary operators.

In the second part of this paper we analyze the RG flow between the $m$’th and
$(m-1)$’th diagonal Virasoro minimal model in AdS. We will do so
perturbatively at large $m$ in order to maintain control. In an upcoming paper
we will analyze the same flow for finite $m$ numerically [2].

As we discuss in more detail in appendix B, the consistent elementary boundary
conditions for the minimal models on the half-space are well-known to be the
Cardy boundary conditions. They are labeled by two integers:

$\textbf{a}\equiv(a_{1},a_{2})_{m},\quad 1\leq a_{1}\leq m-1,\quad 1\leq
a_{2}\leq m,\quad(a_{1},a_{2})\simeq(m-a_{1},m+1-a_{2})\,.$ (1.1)

Each of these allowed elementary boundary conditions is also a consistent
starting point of the RG flow in AdS.

The flow itself is triggered by deforming the bulk theory by the
$\phi_{(1,3)}$ operator. In AdS the integrability of these flows [12] does not
appear to survive, but at large $m$ the flow is under perturbative control
[13] (see also [14] for a review). By all accounts the bulk flow ends in the
$(m-1)$’th minimal model, but the fate of the boundary conditions is less
clear. A central question in this work is how precisely the flow in the space
of boundary conditions on AdS differs from that on the flat half-space.

On the flat half-space the RG flow was analyzed perturbatively in [15]. For
example, for $a_{1},a_{2}\ll m$ and $a_{2}>1$ the authors of [15] found
evidence for a natural flow from $(a_{1},a_{2})_{m}$ to
$(a_{2}-1,a_{1})_{m-1}$. Together with earlier discovered boundary flows
within the $m$’th minimal model [16, 17], there is an interesting web of flows
as shown diagrammatically in figure 2 of [15]. In particular, the diagram
shows that reaching $(a_{2},a_{1})_{m-1}$ from $(a_{1},a_{2})_{m}$ can only be
done through a sequence of flows that passes through two intermediate fixed
points:

$(a_{1},a_{2})_{m}\to\bigoplus_{l=1}^{\min(a_{1},a_{2})}(a_{1}+a_{2}+1-2l,1)_{m}\to\bigoplus_{l=1}^{\min(a_{1},a_{2})}(1,a_{1}+a_{2}+1-2l)_{m-1}\to(a_{2},a_{1})_{m-1}~{},$
(1.2)

where the first and last arrow correspond to the boundary flow of [16, 17]
(with opposite signs) and the second arrow to a bulk flow.

The “boundary follows the bulk” principle means that, in contrast to [15], we
should not have to contend with independent couplings and beta functions for
the boundary operators in AdS. This considerably simplifies our analysis. We
provide a bit of evidence for a much simpler picture and, based on the
anomalous dimensions of boundary operators, speculate that

$(a_{1},a_{2})_{m}\to(a_{2},a_{1})_{m-1}~{},$ (1.3)

is the covariant flow in AdS.

Below we will analyze the perturbed one-point functions of bulk operators and
two-point functions of boundary operators. One peculiar feature is the
appearance of divergences linear in $m$ in certain integrals which are
cancelled by $1/m$ behavior in the (OPE) coefficients that multiply the
integrals. The divergences themselves are entirely analogous to the
$1/\epsilon$ poles in dimensional regularization (and one may call them
‘$1/(1/m)$ poles’). Normally one would add suitable counterterms to
renormalize these divergences, but there is no need to do so if they multiply
suppressed coefficients. This phenomenon however invalidates any estimate of
the magnitude of certain contributions (in the $1/m$ expansion) that is based
purely on OPE coefficients.

#### Material in the appendices

Our perturbative QFT in AdS results are obtained by Weyl rescaling and
integrating correlation functions of two-dimensional BCFTs. These BCFT
correlators warrant a separate discussion, which we present in appendices A
(for general 2d BCFTs) and B (for the minimal models specifically). The
material presented there is a substantial part of the work done for this paper
and contains several new results.111In particular, to the best of our
knowledge the equations (B.3), (B.5), (B.7), (B.9) are all new. We would also
like to highlight equations (B.3.1), (B.52), (B.78), (B.7.1), (B.7.1). These
can in principle be derived using F-matrices, but the latter are usually
difficult to implement for continuous values of the parameters. As such it
could have been included in the main text, but its inclusion as an appendix
hopefully clarifies that it is stand-alone material which might be useful
beyond the QFT in AdS discussion. We advise the reader to at least briefly
consult these appendices before starting with subsection 3.3 and section 4.

#### Ancillary material

With the submission of this work to the arXiv e-print archive we have included
a Mathematica notebook ancillary.nb which contains all the substantial
computations done in this work.

##  2 General principles

In this and the next section we will discuss some general principles of QFT in
AdS as well as one-loop perturbation theory. Many of these principles have
appeared (sometimes implicitly) in the literature to date, but we think that
it is useful to present a systematic viewpoint.

###  2.1 From BCFT to AdS

In Poincaré coordinates the AdS metric reads

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{R^{2}}{z^{2}}(dz^{2}+d\vec{x}^{2})~{},$
(2.1)

and it is immediately seen that AdSd+1 is Weyl equivalent to the half-space
defined as the subspace of flat $\mathbb{R}^{d+1}$ with $z>0$. Therefore any
consistent CFT defined on a half-space, so any BCFT, can immediately be put in
AdS by the usual Weyl rescaling rules. To illustrate this consider for example
the general form of the BCFT one- and two-point functions of scalar conformal
primaries:

$\begin{split}\langle\mathcal{O}(\vec{x},z)\rangle_{\text{BCFT}}&=\frac{B_{\mathcal{O}}}{(2z)^{\Delta}}~{},\\\
\langle\mathcal{O}_{1}(\vec{x}_{1},z_{1})\mathcal{O}_{2}(\vec{x}_{2},z_{2})\rangle_{\text{BCFT}}&=\frac{f(\xi)}{(2z_{1})^{\Delta_{1}}(2z_{2})^{\Delta_{2}}},\qquad\xi=\frac{(\vec{x}_{1}-\vec{x}_{2})^{2}+(z_{1}-z_{2})^{2}}{4z_{1}z_{2}}~{},\end{split}$
(2.2)

with $B_{\mathcal{O}}$ a BCFT-dependent coefficient and $f(\xi)$ a BCFT-
dependent function. For these operators the Weyl rescaling rule is simply

$\langle\mathcal{O}(\vec{x},z)\ldots\rangle_{\text{AdS}}=(z/R)^{\Delta}\langle\mathcal{O}(\vec{x},z)\ldots\rangle_{\text{BCFT}}~{},$
(2.3)

and therefore

$\begin{split}(2R)^{\Delta}\langle\mathcal{O}(\vec{x},z)\rangle_{\text{AdS}}&=B_{\mathcal{O}}\,,\\\
(2R)^{\Delta_{1}+\Delta_{2}}\langle\mathcal{O}_{1}(\vec{x}_{1},z_{1})\mathcal{O}_{2}(\vec{x}_{2},z_{2})\rangle_{\text{AdS}}&=f(\xi)\,.\end{split}$
(2.4)

In words, we see that one-point functions are constant and two-point functions
depend only on the geodesic distance $d_{1,2}$ since

$2\xi+1=\cosh(d_{1,2}/R)\,.$ (2.5)

Let us now draw some more general lessons. The form (2.4) is also the one
required by AdS general covariance, fundamentally because the BCFT conformal
algebra $\mathfrak{so}(d,1)$ is precisely the same as the AdS isometry
algebra. There are in fact no _further_ Ward identities (at fixed $R$) that
follow from conformal invariance of the theory: the form (2.4) is the most
general one for an arbitrary non-conformal QFT in AdS.222This phenomenon
extends to spinning operators. For example, the two-point function of (not
necessarily traceless) spin 2 operators in AdS as computed recently in [7]
essentially agrees with the corresponding BCFT correlator [18]. Therefore one
needs to work a little harder to see that the bulk theory is conformal. One
method is to inspect the undetermined functions like $f(\xi)$ in order to find
out that they are compatible with a bulk conformal OPE. Another is to compare
the theories at different values of $R$. For example, in a general QFT with a
mass scale $\mu$ the one-point function coefficients $B_{\mathcal{O}}$ can be
complicated functions of $\mu R$, whereas in a CFT the $R$-dependence must be
trivial.

The above analysis immediately extends to higher-point functions. Covariant
correlation functions must depend only on the geodesic distances between
points and, for non-conformal theories, on the dimensionless combination $\mu
R$. These dependences are not otherwise constrained by first principles.

In the above we used the dimensionless combinations
$R^{\Delta}\mathcal{O}(\vec{x},z)$. In a general non-conformal theory the bulk
operator may of course obtain a non-trivial scaling, which can however
similarly be offset with powers of $R$. It is convenient to always work with
such a dimensionless version of the bulk operators.

###  2.2 Towards the boundary

Consider now sending the insertion points towards the conformal boundary of
AdS by taking $z\to 0$. For BCFT correlation functions this limit is defined
by the boundary operator expansion. For scalar primaries we can for example
write:

$\langle\mathcal{O}(\vec{x},z)\ldots\rangle_{\text{BCFT}}=\sum_{\hat{\mathcal{O}}}z^{{\hat{\Delta}}_{\hat{\mathcal{O}}}-\Delta_{\mathcal{O}}}B_{\mathcal{O}}^{\phantom{\mathcal{O}}\hat{\mathcal{O}}}\langle\hat{\mathcal{O}}(\vec{x})\ldots\rangle_{\text{BCFT}}~{},$
(2.6)

with $\hat{\mathcal{O}}(\vec{x})$ a boundary operator. The AdS version then
reads:

$R^{\Delta_{\mathcal{O}}}\langle\mathcal{O}(\vec{x},z)\ldots\rangle_{\text{AdS}}=\sum_{\hat{\mathcal{O}}}z^{{\hat{\Delta}}_{\hat{\mathcal{O}}}}B_{\mathcal{O}}^{\phantom{\mathcal{O}}\hat{\mathcal{O}}}\langle\hat{\mathcal{O}}(\vec{x})\ldots\rangle_{\text{AdS}}~{},$
(2.7)

so we Weyl rescale the bulk operator but not the boundary operator. The
boundary operator correlation functions are then unchanged, for example:

$\langle\hat{\mathcal{O}}(\vec{x})\hat{\mathcal{O}}(\vec{y})\rangle_{\text{BCFT}}=\frac{1}{(\vec{x}-\vec{y})^{2\hat{\Delta}_{\hat{\mathcal{O}}}}}=\langle\hat{\mathcal{O}}(\vec{x})\hat{\mathcal{O}}(\vec{y})\rangle_{\text{AdS}}~{},$
(2.8)

and similarly we find the usual conformal structures for any $n$-point
function of boundary operators.

How does this work for a non-conformal QFT in AdS? The general idea,
formulated for example in [9], is that there is still a convergent state-
operator correspondence, or more precisely a correspondence between bulk
states and boundary operators. If the AdS radial Hamiltonian can be
diagonalized and has discrete spectrum then it defines a set of boundary
operators that transform covariantly under conformal transformations. The
expansion as in (2.7) is then fixed by symmetry; one quick way to conclude
this is to notice the functional form of the
$\langle\mathcal{O}(\vec{x},z)\hat{\mathcal{O}}(\vec{y})\rangle$ two-point
function is completely fixed. For a non-conformal field theory the
coefficients $b_{\mathcal{O}}^{\phantom{\mathcal{O}}\hat{\mathcal{O}}}$ and
$\hat{\Delta}$ can depend non-trivially on the dimensionless ratio $\mu R$.

####  2.2.1 Distance versus curvature

Consider a non-conformal field theory in AdS and hold fixed its mass scale
$\mu$. Correlation functions of local bulk operators can have an arbitrary
dependence on invariant distances, say $d(\cdot,\cdot)/R$, and on the
combination $\mu R$. Physically, however, it is not immediately clear what
happens when we consider the near-boundary or large-distance limit of these
correlation functions. Do we always observe the infrared behavior of the
theory, as would be the case for theories in flat space? Or does the curvature
intervene and cut off the RG flow at the scale $\mu R$? It appears to us that
the latter option must be realized. (This viewpoint was also taken in [4]. As
argued in that paper, see also [5], observables do not need to be smooth as a
function of $\mu R$.) To see this we appeal to the boundary operator expansion
which ensures that every correlation function has a convergent boundary block
decomposition. The dimensionless parameters $\hat{\Delta}_{\mathcal{O}}$ and
$b_{\mathcal{O}}^{\phantom{\mathcal{O}}\hat{\mathcal{O}}}$ appearing in this
decomposition are insensitive to any distances, and therefore can only depend
on $\mu R$. So we conclude that, by varying the curvature radius, we should be
able to see the entire RG flow even if we restrict ourselves to boundary
correlation functions (or, more generally, the longest distances in AdS).

##  3 Hyperbolic life at one loop

Suppose we switch on a marginal or relevant deformation in AdS by a local
operator $\mathcal{O}(x)$ with scaling dimension $\Delta_{\mathcal{O}}$. The
correlation functions in the deformed theory can be computed perturbatively by
expanding

$\langle\ldots\exp\left(-gR^{\Delta_{\mathcal{O}}-D}\int
d^{D}x\sqrt{g}\,\mathcal{O}(x)+S_{\text{c.t.}}\right)\rangle~{},$ (3.1)

in the dimensionless coupling $g$. (We do not expect denoting the metric by
$g_{\mu\nu}$ and the coupling by $g$ to cause any confusion; in particular,
the latter will always appear outside integrals.) Rather than introducing a
new dimensionful parameter, we have introduced the naturally available AdS
curvature radius to make the perturbation dimensionless.

The counterterm action in the above expression is necessary to cancel both
bulk and boundary divergences. The bulk divergences are of the UV type. They
are essentially unchanged compared to those in flat space, although new
counterterms involving the AdS curvature may be needed. Their renormalization
is well understood, see for example [11], so we will only briefly consider
them in the next subsection. Afterwards we will focus on the _infrared_
divergences which arise close to the AdS boundary.

###  3.1 Ultraviolet behavior

In this subsection we will consider a nearly marginal deformation, so we set

$\Delta_{\mathcal{O}}=D-\epsilon\,,$ (3.2)

and consider the limit of small $\epsilon$.

Let us first investigate the one-loop running in AdS and in BCFT. To this end
we consider

$\frac{1}{2}g^{2}\mu^{2\epsilon}\int_{x_{1}}\mathcal{O}(x_{1})\int_{x_{2}}\mathcal{O}(x_{2})~{},$
(3.3)

in either setup. We can assume the self-OPE of $\mathcal{O}(x)$ to take the
form

$\mathcal{O}(x)\mathcal{O}(y)=\frac{1}{|x-y|^{2\Delta_{\mathcal{O}}}}+\frac{C_{\mathcal{O}\mathcal{O}\mathcal{O}}\mathcal{O}(y)}{|x-y|^{\Delta_{\mathcal{O}}}}+\ldots~{},$
(3.4)

so the term leading to renormalization of the coupling constant reads

$\frac{1}{2}g^{2}\mu^{2\epsilon}C_{\mathcal{O}\mathcal{O}\mathcal{O}}\int_{x_{2}}\mathcal{O}(x_{2})\times\begin{cases}\int
d^{d}\vec{x}\int_{0}dz_{1}|x_{1}-x_{2}|^{-\Delta_{\mathcal{O}}}&\qquad\text{BCFT}\\\
\int
d^{d}\vec{x}\int_{0}dz_{1}R^{D-\Delta_{\mathcal{O}}}z_{1}^{\Delta_{\mathcal{O}}-D}|x_{1}-x_{2}|^{-\Delta_{\mathcal{O}}}&\qquad\text{AdS}\end{cases}$
(3.5)

with the extra factors in the AdS integral originating from the metric
determinant and the Weyl rescaling. Doing the parallel integral yields

$\frac{1}{2}g^{2}\mu^{2\epsilon}C_{\mathcal{O}\mathcal{O}\mathcal{O}}\frac{\pi^{d/2}\Gamma((\Delta-d)/2)}{\Gamma(\Delta/2)}\int_{x_{2}}\mathcal{O}(x_{2})\times\begin{cases}\int_{0}dz_{1}|z_{1}-z_{2}|^{\epsilon-1}&\qquad\text{BCFT}\\\
\int_{0}dz_{1}R^{\epsilon}z_{1}^{-\epsilon}|z_{1}-z_{2}|^{\epsilon-1}&\qquad\text{AdS}\end{cases}$
(3.6)

The integral over the orthogonal direction diverges, but since we are only
interested in the short-distance singularities we can just cut it off at a
distance $L$ from $z_{2}$. In a more physical setup such an $L$ would more
naturally be provided by another operator insertion, and so one should think
of $L$ as the typical scale of the observable under consideration. With such a
cutoff we find:

$\frac{1}{2\epsilon}g^{2}\mu^{2\epsilon}C_{\mathcal{O}\mathcal{O}\mathcal{O}}\frac{\pi^{d/2}\Gamma((\Delta-d)/2)}{\Gamma(\Delta/2)}\int_{x_{2}}\mathcal{O}(x_{2})\times\begin{cases}L^{\epsilon}&\qquad\text{BCFT}\\\
\begin{cases}L^{\epsilon}\qquad L\ll R\\\ R^{\epsilon}\qquad R\ll
L\end{cases}&\qquad\text{AdS}\end{cases}$ (3.7)

where we ignored non-pole terms in $\epsilon$ — in AdS these are a non-trivial
function of $R/L$ for which we show the dominant term. From this computation
we can first of all conclude that the coupling self-renormalization is the
same in both theories. Secondly, to the extent that we can trust this one-loop
result, we see that in AdS the natural dimensionless combination is $\mu R$
for sufficiently large distances. On the other hand, in BCFT the natural
dimensionless ratio is $\mu L$ (as long as we can ignore boundary effects).
This is again in agreement with the general discussion presented above, where
we argued that in AdS the coupling constants naturally run with the curvature
radius $R$ and not with the size $L$ of the observable. This also provides a
concrete illustration of the picture sketched in the introduction of [5].

Another interesting aspect of renormalization in curved space is the
existence, for every local operator $\mathcal{O}(x)$ of dimension $\Delta$, of
sequences of additional local operators of the schematic form

$(\text{Riemann})^{n}\mathcal{O}(x)~{},$ (3.8)

where $(\text{Riemann})^{n}$ is a degree $n$ homogeneous polynomial in the
Riemann tensor with index contractions chosen at will. These operators will
have dimensions $\Delta+2n$ and it is essential to incorporate them in order
to understand operator mixing in perturbation theory.

This phenomenon arises for example when we consider the integral of the first
term in equation (3.4). Let us use point-splitting to regulate the
corresponding short-distance divergence in the integral. In flat space we then
find a simple power-law divergence $\Lambda^{2\Delta-D}$, which can be
absorbed by including a (divergent) operator mixing with the identity operator
${\mathds{1}}$. In AdS we will of course find the same divergence to leading
order but there can also be subleading terms of the form
$\Lambda^{2\Delta-D-2n}$. If some of these terms are divergent then they need
to be renormalized, and for scalar operators the correct way to do this is by
including operator mixing with the additional operator
$(\text{Ric})^{n}{\mathds{1}}$. Notice finally that, if $\Delta=D-\epsilon$
and $D$ is even then there might in principle appear $1/\epsilon$ poles as
well, but for AdS an explicit computation shows that this is not the case.

Consider now a theory where there exist two operators $\mathcal{O}_{1}(x)$ and
$\mathcal{O}_{2}(x)$ with scaling dimensions such that
$\Delta_{2}=\Delta_{1}+2$ (plus perhaps perturbatively small corrections) and
otherwise the same quantum numbers. In that case there is an ambiguity: for
any real $\alpha$ the operator

$\mathcal{O}_{2}^{\prime}(x)=\mathcal{O}_{2}(x)+\alpha(\text{Ric})(x)\mathcal{O}_{1}(x)~{},$
(3.9)

will be just as good as $\mathcal{O}_{2}(x)$. On a general curved manifold
there is no scheme-independent way to fix such ambiguities, although
exceptions exist in special cases. For example, in [19] the coefficients were
fixed on the round $S^{4}$ by demanding that two-point functions are diagonal
and one-point functions are vanishing. The extra symmetries of AdS do not help
much in this regard, although at a fixed point one can fix $\alpha$ by
demanding that the one-point functions match those of the corresponding BCFT.

###  3.2 Infrared divergences

Infrared divergences in conformal perturbation theory are not uncommon. For
example, in flat space conformal correlation functions fall off like
$x^{-2\Delta}$, so infrared divergences arise whenever $\Delta\leq d/2$. An
example of such a strongly relevant deformation is the mass deformation for
the CFT of a free scalar field. A divergence can already be found at first
order in perturbation theory:

$\delta\langle\phi(0)\phi(y)\rangle=-\mu^{2}\int
d^{D}x\langle\phi(0)\phi(y)\phi^{2}(x)\rangle\approx-\mu^{2}S_{D}\int\frac{dr}{r^{D-3}}=-\mu^{2}S_{D}\frac{R^{4-D}}{4-D}~{},$
(3.10)

where we had to cut off the integral at $|x|<R$ to make it finite. One might
now decide to stop the computation here, because the infrared divergence
indicates that the theory becomes strongly coupled at large distances. This is
however too pessimistic. After all, the flat-space propagator of a massive
free scalar field has a good expansion in $\mu^{2}x^{2}$ which we should be
able to compute perturbatively. And indeed the divergence can easily be
cancelled if we add a boundary counterterm at $|x|=R$. Since

$\frac{\mu^{2}R}{4-D}\int_{|x|=R}d^{D-1}x\langle\phi^{2}(0)\phi^{2}(x)\rangle=\mu^{2}S_{D}\frac{R^{4-D}}{4-D}\,,$
(3.11)

a deformation of the form:

$\langle\ldots\exp\left(-\mu^{2}\int_{|x|\leq
R}d^{D}x\,\phi^{2}(x)+\frac{\mu^{2}R}{4-D}\int_{|x|=R}d^{D-1}x\,\phi^{2}(x)~{}\right)\rangle~{},$
(3.12)

is infrared finite to first order in perturbation theory.

In AdS the situation is richer. Substituting the boundary operator expansion,
we find for small $z$ that

$\int d^{D}x\sqrt{g}\,\mathcal{O}(x)\approx\int
d^{d}\vec{x}\int_{0}dz\,B_{\mathcal{O}\hat{}\mathcal{O}}z^{\hat{\Delta}-d-1}\hat{}\mathcal{O}(\vec{x})~{},$
(3.13)

which is divergent for any marginal or relevant boundary operator with
$\hat{\Delta}\leq d$. We can remove the leading divergence by cutting the $z$
integral off at a small value $a$ and adding a counterterm

$\frac{R}{\hat{\Delta}-d}\int
d^{d}\vec{x}\,\sqrt{h}\,\mathcal{O}(\vec{x},a)~{},$ (3.14)

with $h$ the induced metric at the cutoff surface and an extra $R$ inserted
for dimensional reasons. Notice that the counterterm involves the _bulk_
operator $\mathcal{O}(\vec{x},a)$. Subleading divergences can be removed by
adding other operators or by a counterterm involving normal derivatives, for
example

$\int d^{d}\vec{x}\,\sqrt{h}\,n^{\mu}\partial_{\mu}\mathcal{O}(\vec{x},a)~{},$
(3.15)

and appropriately tuning the coefficients.

When $\hat{\Delta}=d$ the divergence is logarithmic and we need a counterterm
of the form

$\log(a/R)\int d^{d}\vec{x}\,\sqrt{h}\,\mathcal{O}(\vec{x},a)~{},$ (3.16)

where we inserted a factor $R$ on dimensional grounds. This leads to a small
breaking of conformal invariance, which we may capture in terms of a beta
function for the boundary coupling. In general one can however tune this
marginal boundary coupling as a function of the bulk coupling in order to
retain boundary conformal invariance at each order in $\lambda$. This is for
example why the boundary correlation functions for a free scalar around the BF
point, where $\phi^{2}$ is marginal, remain conformally covariant. (See
section 6 of [20] for some more details, and also [6] for a pertinent
example.)

####  3.2.1 Covariance and comparison to BCFT

The near-boundary divergences in conformal perturbation theory in AdS are
similar to those found in BCFT. Indeed, at any loop order the integrands in an
AdS background differ from those in a flat BCFT background only by the
insertion of Weyl factors and the metric determinant. There is however an
important qualitative difference: the AdS divergences are of an _infrared_
nature and the BCFT divergences are of an _ultraviolet_ nature. This
completely changes the way in which the divergences are cancelled by
counterterms.

First, in AdS the counterterms are naturally defined on the cutoff surface in
order to retain covariance. To see this, consider a Killing vector field
$\xi^{\mu}$ in AdS, and consider the invariance of the first-order correction
to any correlation function $\langle\ldots\rangle$ along $\xi$. Concretely
this means that we should impose that transporting every operator in the
perturbed correlation function along $\xi$ by means of the Lie derivative
$\mathcal{L}_{\xi}$ gives zero. Since the bare correlation function is by
assumption invariant, we can move the Lie derivative to the perturbing
operator $\mathcal{O}(x)$, leading to the following demand:

$0=\int_{a}d^{D}x\sqrt{g}\,\xi^{\mu}\nabla_{\mu}\langle\mathcal{O}(x)\ldots\rangle+\mathcal{L}_{\xi}S_{\text{c.t.}}~{},$
(3.17)

where we assumed that $\mathcal{O}(x)$ is a scalar. After integration by parts
we find a boundary term at the cutoff surface:

$0=\left.\int
d^{d}x\sqrt{h}\,\xi^{\mu}n_{\mu}\langle\mathcal{O}(x)\ldots\rangle\right|_{a}+\mathcal{L}_{\xi}S_{\text{c.t.}}\,.$
(3.18)

This is a simple differential equation for the counterterm action. It is most
naturally solved by a counterterm _on_ the near-boundary cutoff surface, which
is why such counterterms are the correct covariant ones.

In BCFT no such constraint arises because all the remaining Killing vector
fields leave the boundary invariant and therefore obey $n_{\mu}\xi^{\mu}=0$.
It is then customary, and indeed much simpler, to define the counterterm
action on the actual boundary rather than at the cutoff surface. Such a
procedure is of course entirely in keeping with the usual way that UV
divergences are renormalized. For example, consider the point-splitting
regularization procedure to renormalize a bare local operator
$\mathcal{O}_{i}(x)$. The renormalization is always done by replacing the bare
operator by a renormalized operator $Z_{i}^{\phantom{i}j}\mathcal{O}_{j}(x)$,
which in particular is inserted at exactly the same point as the original
operator rather than, say, integrated along the cutoff surface.

As we have seen, demanding general covariance also means that the boundary
counterterms are _completely determined_ by the bulk RG flow. In BQFT it is
well known that there can be independent bulk and boundary RG flows; in
particular, the boundary can undergo an RG flow while the bulk theory remains
critical. In AdS/CFT a similar phenomenon occurs for example in the form of
the double-trace flow [21], where a double-trace deformation of the CFT is
encoded holographically by a changing of the boundary conditions of the bulk
fields. However such flows are unnatural in the precise sense that they do not
retain AdS covariance. If we insist that the deformed correlation functions
remain covariant (and if there is no exactly marginal boundary coupling) then
there is no freedom left to apply independent boundary deformations. In this
way we can say that “the boundary follows the bulk” for QFT in AdS.

####  3.2.2 Comparison to holographic renormalization

The importance of the covariance of the counterterms is familiar from previous
works on holographic renormalization in the gauge/gravity dualities; see [22]
for an introduction. Most of that work considers classical gravity with matter
on AAdS backgrounds, although recently loop corrections were also considered
more systematically in [23]. In a gravity setup there are no local bulk
operators and it is more natural to consider the partition function as a
functional of the boundary conditions $\phi_{(0)}$ for the bulk fields. In the
modern understanding the one-point function of the boundary operator is then
given by the (renormalized) conjugate radial momentum of the bulk field. The
renormalization of the divergences in the on-shell action leads to an infamous
factor $2\Delta-d$ [24] compared to a more naive prescription.

We do not consider theories of gravity. Therefore we do have local bulk
operators and we can simply consider their limit as we send them to the
boundary. One immediately verifies [25] that this leads to the same
normalization for two-point functions, including the extra factor $2\Delta-d$,
because of the difference between the bulk-bulk and bulk-boundary propagator:

$G_{\text{bu-
bu}}(x,x^{\prime})=z^{\Delta}\left(\frac{1}{2\Delta-d}G_{\text{bu-
bou}}(\vec{x},x^{\prime})+O(z)\right)~{},$ (3.19)

holding $x^{\prime}$ fixed in the bulk. Although this prescription suffices
for our purposes, it does not give all the same information as the more
familiar holographic renormalization prescription. In particular, since the
boundary correlation functions are never integrated (against sources) we are
insensitive to their potentially ill-defined distributional nature.
Correspondingly it is not straightforward to see contact terms in Ward
identities of boundary currents (including a potential stress tensor) or their
anomalies [26]. When such questions are important for QFT in AdS we expect
that a more refined analysis will be needed. It is then a matter of taste
whether to take the more abstract viewpoint of conformal perturbation theory,
as we do in this work, or Lagrangian perturbation theory, as was for example
done in [23].333There is one more sense in which the renormalization of the
on-shell action, or partition function at loop level, is richer than that of
the correlation functions. According to [27] a specific boundary action is
needed in order to have a well-defined variational principle for the bulk
fields. It is unclear to us whether this condition can be understood from the
conformal perturbation theory viewpoint that we take here.

####  3.2.3 Is AdS a box?

A massive particle cannot reach the conformal boundary of AdS by moving along
a geodesic, even though part of that boundary is certainly in causal contact
with it. In this sense AdS confines, and can be seen as a box. On the other
hand, for massless particles AdS is much less box-like since they can reach
the boundary in finite global time.

Even for gapped theories, however, it is not at all immediate that (conformal)
perturbation theory in an AdS background is _automatically_ infrared finite.
The earliest claim that it is appears to date from [3]. In that paper the
authors claim that “perturbation theory diagrams are infrared well-behaved”
based on the exponential falloff of bulk-bulk propagators with distance.
However in actuality this exponential falloff can be offset by the
exponentially growing area of spheres in hyperbolic space; a fact that is
actually mentioned a little earlier in the same paper. As we have just seen,
whether a perturbative computation is infrared finite or not depends on the
spectrum of boundary operators, and there is certainly the possibility of
infrared divergences. We are led to conclude that AdS is perhaps not that
great of a box.

One major advantage nevertheless appears to remain: after renormalization the
boundary correlation functions are likely unambiguously defined for massless
and massive theories alike, even when the corresponding massless S-matrix
elements are infrared divergent.

###  3.3 Anomalous dimensions of boundary operators

It is interesting to consider the general first-order correction to two-point
functions of boundary operators under a bulk deformation. Our goal will be to
demonstrate that these correlation functions remain conformal at first order,
and that for equal operators there generically appears a logarithmic
divergence that gives rise to an anomalous dimension.

To first order the two-point function is given by:

$\langle{\hat{\psi}}_{1}(\vec{x}_{1}){\hat{\psi}}_{2}(\vec{x}_{2})\rangle=\frac{\delta_{12}}{{x}_{12}^{2{\hat{\Delta}}_{1}}}-gR^{\Delta-D}\int
d^{D}{x}\sqrt{g}\langle\phi(x){\hat{\psi}}_{1}(\vec{x}_{1}){\hat{\psi}}_{2}(\vec{x}_{2})\rangle+(\text{counterterms})+O(g^{2})~{},$
(3.20)

for a generic deformation by a bulk operator denoted $\phi(x)$ with dimension
$\Delta$. Notice that $\vec{x}_{1}$ and $\vec{x}_{2}$ are boundary points,
$x_{12}\equiv|\vec{x}_{1}-\vec{x}_{2}|$, and $x$ is a bulk point. We will work
in Poincaré coordinates where $z$ is the transverse coordinate. The
dimensionless bulk coupling is denoted $g$.

The boundary counterterms will consist of two parts. First there will be the
generic boundary counterterms discussed in the previous section, which will
cancel any divergences arising as $\phi({x})$ approaches the boundary away
from the insertion points of the boundary operators. Then there will be wave
function renormalization factors that will cancel additional divergences that
appear as $\phi(x)$ approaches ${\hat{\psi}}(\vec{x}_{1})$ or
${\hat{\psi}}(\vec{x}_{2})$.

The three-point function in equation (3.20) can be decomposed into conformal
blocks:

$\displaystyle
x_{12}^{\hat{\Delta}_{1}+\hat{\Delta}_{2}}(2R)^{\Delta}\langle{\hat{\psi}}_{1}({\vec{x}}_{1}){\hat{\psi}}_{2}({\vec{x}}_{2})\phi(\vec{x},z)\rangle=\left(\frac{z^{2}+(\vec{x}_{2}-\vec{x})^{2}}{z^{2}+(\vec{x}_{1}-\vec{x})^{2}}\right)^{{\hat{\Delta}_{12}/2}}\sum_{k}B_{\phi}^{\phantom{\phi}k}\hat{C}_{k{\hat{\psi}}{\hat{\psi}}}f(\hat{\Delta}_{12},\hat{\Delta}_{k},v)~{},$
(3.21)

where $\hat{\Delta}_{12}\equiv\hat{\Delta}_{1}-\hat{\Delta}_{2}$ and the Weyl
rescaling from BCFT to AdS means that we replace a factor $(2z)^{-\Delta}$
with $(2R)^{-\Delta}$. The cross-ratio $v$ is defined as

$\displaystyle
v=\frac{z^{2}x_{12}^{2}}{(z^{2}+({\vec{x}}_{1}-{\vec{x}})^{2})(z^{2}+({\vec{x}}_{2}-{\vec{x}})^{2})}~{}.$
(3.22)

The conformal blocks were computed in [28, 29] and in our conventions read:

$\displaystyle
f(\hat{\Delta}_{12},\hat{\Delta}_{k},v)=(4v)^{\hat{\Delta}_{k}/2}\,_{2}F_{1}\left(\frac{\hat{\Delta}_{k}+\hat{\Delta}_{12}}{2},\frac{\hat{\Delta}_{k}-\hat{\Delta}_{12}}{2};1+\hat{\Delta}_{k}-\frac{d}{2};v\right)~{}.$
(3.23)

This conformal block decomposition converges everywhere in AdS except on the
geodesic that connects $\vec{x}_{1}$ and $\vec{x}_{2}$. Our approach will be
to swap the conformal block decomposition with the integral and analyze the
resulting expressions.

####  3.3.1 Identical operators

We will first focus on identical boundary operators with dimension
$\hat{\Delta}_{1}$. This requires us to integrate:

$\displaystyle G_{k}({x}_{12})\equiv\int
d^{d}\vec{x}\int_{a}^{\infty}\frac{dz}{z^{d+1}}f(0,\hat{\Delta}_{k},v)~{},$
(3.24)

where we introduced a near-boundary cutoff $a$. To do this integral we power-
expand $f(0,\hat{\Delta}_{k},v)$ around $v=0$ and perform the integration
term-wise using equations (C) and (C.10) in appendix C. The divergences that
we find as we send $a\to 0$ are now of two types.

First, for boundary operators with $\hat{\Delta}_{k}<d$ we find a power-law
divergence that has exactly the form predicted by the near-boundary operator
expansion discussed in the previous section. If such operators are present
then we must add the corresponding counterterms to the ‘counterterm’
expression in (3.20). It is however easy to verify that these counterterms do
not give any finite contributions and therefore we can (and will) ignore these
divergences in what follows.

Secondly we find a universal _logarithmic_ divergence which is present for all
values of $\Delta$, $\hat{\Delta}_{1}$ and $\hat{\Delta}_{k}\neq 0$. More
precisely the expression we find is:

$G_{k}(\vec{x}_{12})=X_{k}\log({x}_{12}^{2}/a^{2})+Y_{k}~{},$ (3.25)

with

$\displaystyle X_{k}$
$\displaystyle=-2^{\hat{\Delta}_{k}-\Delta}\pi^{d/2}\frac{\hat{\Delta}_{k}\Gamma\left(1-\frac{d}{2}+\hat{\Delta}_{k}\right)}{(d-\hat{\Delta}_{k})\Gamma\left(1+\frac{\hat{\Delta}_{k}}{2}\right)^{2}}~{},$
(3.26) $\displaystyle Y_{k}$
$\displaystyle=-2^{\hat{\Delta}_{k}-\Delta}\sum_{m=0}^{\infty}\frac{\pi^{d/2}\left(\frac{\hat{\Delta}_{k}}{2}\right)_{m}^{2}\Gamma\left(m+\frac{\hat{\Delta}_{k}}{2}-\frac{d}{2}\right)}{m!\Gamma\left(m+\frac{\hat{\Delta}_{k}}{2}\right)\left(1-\frac{d}{2}+\hat{\Delta}_{k}\right)_{m}}\left(\psi\left(m+\frac{\hat{\Delta}_{k}}{2}\right)-\psi\left(m+\frac{\hat{\Delta}_{k}}{2}-\frac{d}{2}\right)\right)~{},$

with $\psi(n)$ the digamma function. The logarithmic divergence was not
present in our previous near-boundary analysis and must therefore be due to
the presence of the boundary operators. It can be cancelled with a wave
function renormalization factor. If we sum over all the blocks:

$\displaystyle\begin{split}\delta\hat{\Delta}&\equiv\sum_{k\neq{\mathds{1}}}B_{\phi}^{\phantom{\phi}k}\hat{C}_{k{\hat{\psi}}{\hat{\psi}}}X_{k}\,,\\\
\delta\hat{C}&\equiv\sum_{k\neq{\mathds{1}}}B_{\phi}^{\phantom{\phi}k}\hat{C}_{k{\hat{\psi}}{\hat{\psi}}}Y_{k}\,,\end{split}$
(3.28)

then the addition of the wave function renormalization factor

$\displaystyle
Z_{{\hat{\psi}}}=1-\frac{1}{2}g\,\delta\hat{\Delta}\log\left(a^{2}/R^{2}\right)+O(g^{2})~{},$
(3.29)

produces the finite one-loop result:

$\displaystyle({{x}}_{12})^{2\hat{\Delta}_{1}}\langle{\hat{\psi}}^{R}_{1}({\vec{x}}_{1}){\hat{\psi}}^{R}_{1}({\vec{x}}_{2})\rangle=1-g\left(\delta\hat{\Delta}\log({x}_{12}^{2}/R^{2})+\delta\hat{C}\right)+O(g^{2})~{}.$
(3.30)

As expected, this expression is compatible with boundary conformal covariance,
and we read off that $\delta\hat{\Delta}$ is exactly the one-loop anomalous
dimension of the boundary operator.

One remaining question concerns the convergence of the sums in (3.28). There
is no a priori reason that they are convergent, since our integration domain
includes a line where the sum over conformal blocks diverges. In the next
section we will show how the sum is in fact divergent in certain examples.
Fortunately the right answer can be recovered easily with a simple
regularization procedure.

Recently the authors of [7] have also derived a non-perturbative expression
for the scaling dimension of ${\hat{\psi}}$. Their derivation uses special
properties of the bulk stress tensor. It is thereby different from ours, but
their ‘naive’ expression agrees with the one presented here when expanded to
first order in the coupling. (To see this one uses that the trace of the
stress tensor is proportional to the deforming operator.) In contrast with the
more simplistic regularizations that we use below, the paper [7] also provides
a first-principles derivation of an absolutely convergent modification of this
sum rule. The essential ingredients of this derivation are the recently
derived local blocks of [30] for the decomposition of the three-point
functions.

####  3.3.2 Example: anomalous dimension of D in $\text{AdS}_{2}$

As an example, we will take a CFT in $\text{AdS}_{2}$ and compute the first-
order anomalous dimension of the displacement operator D under a deformation
by an operator $\phi$ which we can take to be a generic scalar bulk Virasoro
primary. We recall that D is a universal (global) boundary primary operator
with scaling dimension $\Delta_{\text{D}}=2$. Below we will make essential use
of several results presented in appendices A.2 and A.3, which the reader
should probably consult at this stage.

As discussed in appendix A.3.5, the boundary modes exchanged in the connected
part of $\langle\text{D}\text{D}\phi\rangle$ have
$\hat{\Delta}_{k}=k=2\mathbb{N}^{>0}$ and

$\displaystyle
B_{\phi}^{\phantom{\phi}k}\hat{C}_{k\text{D}\text{D}}=B_{\phi}\frac{\sqrt{\pi}\Delta
e^{\frac{i\pi
k}{2}}(\Delta(k-2)(k+1)+4)\Gamma(k)}{2^{2k-1}\Gamma\left(k-\frac{1}{2}\right)}~{}.$
(3.31)

According to the previous discussion, the anomalous dimension of D should read
(taking into account the non-unit normalization of D)

$\displaystyle\delta\Delta_{\text{D}}=\frac{2\pi\Delta}{c}\frac{B_{\phi}}{2^{\Delta}}\sum_{k=2,4,\dots}S_{k}~{},$
(3.32)

where

$\displaystyle S_{k}=2^{-k}e^{\frac{i\pi
k}{2}}\frac{k(2k-1)(\Delta(k-2)(k+1)+4)\Gamma(k-1)}{\Gamma\left(\frac{k}{2}+1\right)^{2}}~{}.$
(3.33)

Since $S_{k}\sim k^{1/3}$ as $k\rightarrow\infty$, this sum is divergent
because of operators with high scaling dimensions. A simple way of
regularizing it is to define

$\displaystyle S(L)=\sum_{k=2,4,\dots}S_{k}e^{-kL}~{}.$ (3.34)

After performing the sum we can take the $L\rightarrow 0$ limit of the
expansion of $S(L)$ to find

$\displaystyle\delta\Delta_{\text{D}}=\frac{2\sqrt{\pi}\Delta}{c}\frac{B_{\phi}}{2^{\Delta}}\lim_{L\rightarrow
0}S(L)=\frac{4\pi}{c}\frac{B_{\phi}}{2^{\Delta}}(\Delta-2)\Delta~{}.$ (3.35)

This is the correct result: one would obtain the same answer after integrating
the correlator of eq. (A.77) over $\text{AdS}_{2}$.

We can likewise compute the anomalous dimension of $\text{D}^{2}$, i.e. the
first (global) boundary primary after D in the $\text{D}\times\text{D}$ OPE.
The relevant correlation function is discussed in appendix A.3.6, and in
particular equation (A.87) informs us that
$\hat{\Delta}_{k}=k=2\mathbb{N}^{>0}$ and that

$\displaystyle
B_{\phi}^{\phantom{\phi}k}\hat{C}_{k\text{D}^{2}\text{D}^{2}}=B_{\phi}\left(c_{k}^{(1)}\Delta+c_{k}^{(2)}\Delta^{2}+c_{k}^{(3)}\Delta^{3}+c_{k}^{(4)}\Delta^{4}\right)~{},$
(3.36)

where the $c_{k}$’s are defined in eq. (A.3.6). The anomalous dimension of
$\text{D}^{2}$ then reads (again we divided by the normalization of
$\text{D}^{2}$)

$\displaystyle\delta\Delta_{\text{D}^{2}}=\frac{B_{\phi}}{2^{\Delta}}\frac{10\sqrt{\pi}}{c(5c+22)}\sum_{k=2,4,...}S_{k}~{},$
(3.37)

where

$\displaystyle
S_{k}=\frac{2^{k}k\Gamma\left(k+\frac{1}{2}\right)}{(k-1)\Gamma\left(\frac{k}{2}+1\right)^{2}}\left(c_{k}^{(1)}\Delta+c_{k}^{(2)}\Delta^{2}+c_{k}^{(3)}\Delta^{3}+c_{k}^{(4)}\Delta^{4}\right)~{}.$
(3.38)

The sum is again divergent due to boundary operators with high scaling
dimensions. As before, we can regularize it by defining

$\displaystyle S(L)=\sum_{k=2,4,\dots}S_{k}e^{-kL}~{},$ (3.39)

and $\delta\Delta_{\text{D}^{2}}$ can be recovered by taking the $L\rightarrow
0$ of $S(L)$, to find

$\displaystyle\delta\Delta_{\text{D}^{2}}=\frac{B_{\phi}}{2^{\Delta}}\frac{2\pi\Delta(\Delta-2)(20c+25(\Delta-2)\Delta+64)}{c(5c+22)}~{}.$
(3.40)

This is once more the correct result, as we have checked by integrating the
correlator of eq. (A.85) over $\text{AdS}_{2}$.

####  3.3.3 Boundary operators mixing at one-loop

Let us also consider a boundary two-point function of different boundary
primary operators. These must vanish if the bulk RG preserves the AdS
isometries:

$\displaystyle\langle{\hat{\psi}}^{R}_{1}(\vec{x}_{1}){\hat{\psi}}^{R}_{2}(\vec{x}_{2})\rangle=0~{}.$
(3.41)

We will now check that the one-loop correction indeed vanishes. The integral
to perform is now:

$\displaystyle\mathcal{I}_{1,k}$ $\displaystyle=\int
d^{d}{\vec{x}}\int_{a}^{\infty}\frac{dz}{z^{d+1}}~{}\frac{f(\hat{\Delta}_{12},\hat{\Delta}_{k},v)}{(z^{2}+(\vec{x}_{1}-\vec{x})^{2})^{{\hat{\Delta}_{12}/2}}(z^{2}+(\vec{x}_{2}-\vec{x})^{2})^{{\hat{\Delta}_{21}/2}}}~{}.$
(3.42)

As before, we power-expand each block around $v=0$ and, performing the
integration term by term using the contour integral representation for the
hypergeometric functions, we arrive at ($u\equiv a/|\vec{x}_{12}|$)

$\displaystyle\mathcal{I}_{1,k}$
$\displaystyle=\frac{\pi^{d/2}2^{\hat{\Delta}_{k}-\Delta}\Gamma\left(1-\frac{d}{2}+\hat{\Delta}_{k}\right)}{\Gamma\left(\frac{\hat{\Delta}_{k}-\hat{\Delta}_{12}}{2}\right)\Gamma\left(\frac{\hat{\Delta}_{12}+\hat{\Delta}_{k}}{2}\right)}|\vec{x}_{12}|^{-\hat{\Delta}_{1}-\hat{\Delta}_{2}}\int_{-i\infty}^{i\infty}\frac{ds}{2\pi
i}\sum_{m=0}^{\infty}\,u^{2s-2m-{\hat{\Delta}_{k}}{}}$ $\displaystyle\times$
$\displaystyle\frac{\Gamma(s)\Gamma\left(2m-s+\hat{\Delta}_{k}-\frac{d}{2}\right)\Gamma\left(m-s+\frac{\hat{\Delta}_{k}}{2}-{\hat{\Delta}_{12}/2}\right)\Gamma\left(m-s+\frac{\hat{\Delta}_{k}}{2}+{\hat{\Delta}_{12}/2}\right)}{m!\,\Gamma\left(m+\hat{\Delta}_{k}-\frac{d}{2}+1\right)\Gamma(2m-2s+\hat{\Delta}_{k}+1)}~{}.$

We want to investigate the small $u$ behavior of the expressions above. If we
close the contour to the right there are three series of poles that contribute
to the integral:

  * I.

for $-\frac{\hat{\Delta}_{12}}{2}+\frac{\hat{\Delta}_{k}}{2}+m-s=-i$,
$i\in\mathbb{N}$, corresponding to $u^{2i-\hat{\Delta}_{12}}$ powers;

  * II.

for $\hat{\Delta}_{k}+2m-\frac{d}{2}-s=-i$, $i\in\mathbb{N}$, corresponding to
$u^{2m+2i+\hat{\Delta}_{k}-d}$ powers;

  * III.

for $\frac{\hat{\Delta}_{12}}{2}+\frac{\hat{\Delta}_{k}}{2}+m-s=-i$,
$i\in\mathbb{N}$, corresponding to $u^{2i+\hat{\Delta}_{12}}$ powers.

We recognize that the second series gives rise to the familiar infrared
divergences that arise only for $\hat{\Delta}_{k}<d$. Their renormalization
was described above and they do not give rise to finite terms. Depending on
the sign of $\hat{\Delta}_{12}$ we however also find that either the type I or
the type III poles have at least one divergent term.444When
$\hat{\Delta}_{12}=0$ both type I and type III poles contribute and generate
the $\log(u^{2})$ divergence associated to the anomalous dimension of the
external operators. Let us consider for concreteness $0<\hat{\Delta}_{12}<2$.
In this case we let

$\displaystyle\begin{pmatrix}{\hat{\psi}}^{R}_{1}\\\
{\hat{\psi}}^{R}_{2}\end{pmatrix}=\begin{pmatrix}Z_{1}&Z_{12}\\\
Z_{21}&Z_{2}\end{pmatrix}\begin{pmatrix}{\hat{\psi}}_{1}\\\
{\hat{\psi}}_{2}\end{pmatrix}~{},$ (3.44)

with $Z_{1,2}$ the wave-function renormalizations (eq. (3.29)). In order to
remove the leading $a^{-\hat{\Delta}_{12}}$ we need to set:

$\displaystyle
Z_{12}=g\,a^{-\hat{\Delta}_{12}}\sum_{k}B_{\phi}{}^{k}{\hat{C}}_{12k}\delta
Z_{12,k}+O(g^{2})~{},\quad Z_{21}=0.$ (3.45)

The quantity $\delta Z_{12,k}$ is simply determined by the most divergent
power in $\mathcal{I}_{1,k}$, that is the residue of the first pole in the I
series. Once again, the removal of this power-law divergence does not produce
any finite terms and we find that the renormalized two-point function of
unequal operators also vanishes at one loop:

$\langle{\hat{\psi}}^{R}_{1}(\vec{x}_{1}){\hat{\psi}}^{R}_{2}(\vec{x}_{2})\rangle=O(g^{2})\,,$
(3.46)

as expected from conformal covariance. Notice that the additional divergences
that arise for highly unequal operators can be cancelled by introducing a
larger mixing matrix that also includes descendants.

##  4 Flows between minimal models

Our starting point is the $m$’th unitary and diagonal minimal model
$\mathcal{M}_{m+1,m}$, with conformal boundary condition ${\bf
a}=(a_{1},a_{2})_{m}$, on an $\text{AdS}_{2}$ background. As per our general
discussion, its correlation functions are simply the Weyl rescaled correlation
functions of the BCFT setup. In the following we suppose the reader is
familiar with the material in appendix B where we discuss these minimal models
and their conformal boundary conditions.

Consider now deforming this theory by the operator $\phi\equiv\phi_{(1,3)}$.
Perturbed correlation functions take the form:

$\displaystyle\langle\ldots\rangle_{g}=\langle\ldots\exp\left(-gR^{\Delta-2}\int
d^{2}x\sqrt{g}\,\phi(x)+S_{\text{c.t.}}\right)\rangle_{0},$ (4.1)

where the counterterm action is to be defined and the coupling $g$ is seen to
be dimensionless if we recall the UV scaling dimension:

$\displaystyle\Delta=2h_{1,3}=2\frac{m-1}{m+1}=2-4/m+O(m^{-2})~{}.$ (4.2)

The one-loop beta function of $g$ reads:

$\displaystyle\dot{g}=(2-\Delta)g-\pi\,C_{\phi\phi\phi}g^{2}+O(g^{3})~{}.$
(4.3)

Using that $C_{\phi\phi\phi}={4}/{\sqrt{3}}$, we find a fixed point at

$\displaystyle g_{*}=\frac{\sqrt{3}}{\pi m}+O(m^{-2})~{}.$ (4.4)

This perturbative calculation is reliable at large $m$, so when $c$ is close
to $1$. Notice that the presence of the AdS background does not complicate the
above analysis. In particular, as we already discussed in subsection 3.1, at
one loop there is no need to include mixing with the identity operator times
the Ricci scalar (even though it has approximately the same dimension as
$\phi_{(1,3)}$).

It is well-known that the fixed point we have found corresponds to the
$(m-1)$’th minimal model, and this is simply the perturbatively controllable
limit of the famous ‘staircase’ RG flow between consecutive minimal models
[13]. It is perhaps less well-known that the phase portrait for this flow on
the upper half-plane is significantly richer, because the boundary can also
undergo flows that are independent of the bulk. The perturbative picture was
comprehensively analyzed in [15] and discussed in the introduction. We will
now investigate how it gets modified in an AdS background.

###  4.1 Infrared divergences

The near-boundary analysis follows the logic of subsection 3.2. To analyze the
infrared divergences at one loop we are led to consider

$-gR^{\Delta-2}\int_{a}^{b}dy\int
dx\sqrt{g}\langle\phi(x)\ldots\rangle^{\text{connected}}_{\text{AdS}_{2}}~{},$
(4.5)

as the infrared cutoff $a$ goes to zero. Notice that we also introduced $b$,
which we choose such that $z>b$ for all the other insertions. This allows us
to focus on the infrared divergences alone.

The boundary operator expansion of $\phi$ in the BCFT was given in equation
(B.2) and, after subtracting the disconnected piece, contains one (marginally)
relevant boundary operator. Adding the Weyl rescaling factors for AdS we find:

$\phi(x,y)-\langle\phi(x,y)\rangle=B_{(1,3)}^{{\bf
a}\,(1,3)}(2R)^{-\Delta}(2y)^{h_{1,3}}\psi_{(1,3)}(x)+\ldots$ (4.6)

and therefore our target expression equals

$-g\,2^{-h_{1,3}}B_{(1,3)}^{{\bf
a}\,(1,3)}\left(\frac{b^{h_{1,3}-1}}{h_{1,3}-1}-\frac{a^{h_{1,3}-1}}{h_{1,3}-1}\right)\int
dx\langle\psi_{(1,3)}(x)\ldots\rangle_{\text{AdS}_{2}}~{}.$ (4.7)

Since $h_{1,3}<1$, we find a power-law infrared divergence as $a\to 0$. In
accordance with our general discussion, we introduce a counterterm of the
form:

$-\frac{gR^{\Delta-1}}{h_{1,3}-1}\left.\int
dx\sqrt{h}\phi(x)\right|_{y=a}~{},$ (4.8)

which indeed, using $\sqrt{h}=R/y$, cancels the divergence when added to the
bulk integral.

####  4.1.1 Large $m$ limit

So far our analysis of the infrared divergences was merely a repetition of our
general discussion. However as $m\to\infty$ we find that $h_{1,3}\to 1$ and
the nearly marginal boundary operator seemingly produces an extra divergence,
namely a pole multiplying the residual finite piece proportional to
$b^{h_{1,3}-1}$. However for the minimal models this divergence is cancelled
by the behavior of the bulk-boundary OPE coefficient: from equation (B.3.1) we
find that

$B_{(1,3)}^{{\bf a}\,(1,3)}\propto\frac{1}{m}~{},$ (4.9)

as $m\to\infty$. So the whole expression actually remains finite555Notice that
we are assuming that the correlator $\langle\psi(x)\ldots\rangle$ remains
finite in the large $m$ limit. If this is not the case our analysis will need
to be modified. and there is no new divergence!

We would like to stress that we introduce here the following general
principle: an integral that diverges as $m\to\infty$ does not need to be
renormalized if it is multiplied by an OPE coefficient that cancels that
divergence. We believe this is justified, and in any case we obtain manifestly
finite answers in the end which, absent some hidden inconsistency, need to be
given a physical interpretation. It would be interesting to understand this
phenomenon better, and perhaps verify whether similar issues arise in the
flat-space BQFT analysis.

###  4.2 Anomalous dimensions of boundary operators

We now turn to the one-loop computation of the two-point function of boundary
operators.

####  4.2.1 Spectrum of boundary Virasoro primaries

Let us consider a generic conformal boundary condition ${\bf
a}=(a_{1},a_{2})_{m}$ that supports a boundary Virasoro primary $\psi_{(r,s)}$
with scaling dimension

$\displaystyle\hat{\Delta}_{r,s}=h_{r,s}=\frac{\bigl{(}(m+1)r-ms\bigr{)}^{2}-1}{4m(m+1)}~{}.$
(4.10)

The one-loop correction to the two-point correlation function of
$\psi_{(r,s)}$ is then

$\displaystyle R^{-\Delta}G_{1}(x_{12})$
$\displaystyle\equiv\int_{-\infty}^{\infty}d{}x\int_{a}^{\infty}\frac{d{}y}{y^{2}}~{}\langle\psi_{(r,s)}(x_{1})\psi_{(r,s)}(x_{2})\phi(x+iy,x-iy)\rangle^{c}$
$\displaystyle-\frac{1}{1-\Delta/2}\int_{-\infty}^{\infty}\frac{d{}x}{a}~{}\langle\psi_{(r,s)}(x_{1})\psi_{(r,s)}(x_{2})\phi(x+ia,x-ia)\rangle^{c}~{}.$
(4.11)

The correlation function in the integrands above is666This can be obtained
from eq. (B.5), upon taking the connected contribution and going to
$\text{AdS}_{2}$ via a Weyl rescaling.

$\displaystyle\langle\psi_{(r,s)}(x_{1})\psi_{(r,s)}(x_{2})\phi(x+iy,x-iy)\rangle^{c}=\frac{\tilde{\mathcal{G}}(\tilde{\eta})-\alpha_{(1,1)}}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}(2R)^{\Delta}}~{},$
(4.12)

where $\tilde{\mathcal{G}}(\tilde{\eta})$ is a sum of three Virasoro blocks

$\displaystyle\tilde{\mathcal{G}}(\tilde{\eta})=\alpha_{(1,1)}V_{(1,1)}(\tilde{\eta})+\alpha_{(1,3)}V_{(1,3)}(\tilde{\eta})+\alpha_{(1,5)}V_{(1,5)}(\tilde{\eta})~{},$
(4.13)

which, as shown in appendix B.5, read (here and below $q\equiv
s-r\in\mathbb{Z}$)

$\displaystyle V_{(1,1)}(\tilde{\eta})$
$\displaystyle=\,_{3}F_{2}\left(\frac{1-m}{m+1},\frac{1+mq-r}{m+1},\frac{1-mq+r}{m+1};\frac{2-m}{m+1},\frac{m+3}{2m+2};\frac{\tilde{\eta}}{4}\right)~{},$
$\displaystyle V_{(1,3)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{h_{1,3}/2}\,_{3}F_{2}\left(\frac{1-m}{2m+2},\frac{m+2mq-2r+1}{2m+2},\frac{m-2mq+2r+1}{2m+2};\frac{3-m}{2m+2},\frac{3m+1}{2m+2};\frac{\tilde{\eta}}{4}\right),$
$\displaystyle V_{(1,5)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{h_{1,5}/2}\,_{3}F_{2}\left(\frac{m}{m+1},\frac{2m+mq-r}{m+1},\frac{2m-mq+r}{m+1};\frac{3m}{m+1},\frac{5m+1}{2m+2};\frac{\tilde{\eta}}{4}\right)~{},$
(4.14)

with cross-ratio $\tilde{\eta}$ defined as:

$\displaystyle\tilde{\eta}=\frac{4y^{2}(x_{12})^{2}}{\left((x_{1}-x)^{2}+y^{2}\right)\left((x_{2}-x)^{2}+y^{2}\right)}~{},\quad
0\leq\tilde{\eta}\leq 4~{}.$ (4.15)

The coefficients in (4.12) and (4.13) are

$\displaystyle\alpha_{(1,1)}\equiv B_{(1,3)}^{{\bf
a}\,(1,1)}~{},\quad\alpha_{(1,3)}\equiv{\hat{C}}_{(r,s)(r,s)(1,3)}^{{\bf
a}}B_{(1,3)}^{{\bf
a}\,(1,3)}~{},\quad\alpha_{(1,5)}\equiv{\hat{C}}_{(r,s)(r,s)(1,5)}^{{\bf
a}}B_{(1,3)}^{{\bf a}\,(1,5)}~{},$ (4.16)

with $B_{(1,3)}^{{\bf a}\,(1,1)}$, $B_{(1,3)}^{{\bf a}\,(1,3)}$, and
$B_{(1,3)}^{{\bf a}\,(1,5)}$ the bulk-boundary coefficients discussed in
detail in appendix B.

##### Computation at finite $m$

In order to compute the integrals in eq. (4.2.1) it is convenient to discuss
separately the various contributions: from $V_{(1,1)}$, $V_{(1,3)}$ and
$V_{(1,5)}$. Let us start from the $V_{(1,1)}$ contribution, which from eq.
(4.2.1) reads

$\displaystyle\alpha_{(1,1)}\int_{-\infty}^{\infty}d{}x\int_{a}^{\infty}\frac{d{}y}{y^{2}}~{}\frac{V_{(1,1)}(\tilde{\eta})-1}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}-\frac{\alpha_{(1,1)}}{1-\Delta/2}\int_{-\infty}^{\infty}\frac{d{}x}{a}~{}\frac{V_{(1,1)}(\tilde{\eta})-1}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}\bigg{\rvert}_{y=a}~{}.$
(4.17)

To compute these integrals we write $V_{(1,1)}$ as a power-series expansion
around $\tilde{\eta}=0$ and perform the integration term by term using the
integration rules of appendix C. Upon performing the infinite sums and
neglecting terms that vanish as we send $a\rightarrow 0$ we find

$\displaystyle\mathcal{I}_{(1,1)}\equiv$
$\displaystyle\alpha_{(1,1)}\int_{-\infty}^{\infty}d{}x\int_{a}^{\infty}\frac{d{}y}{y^{2}}~{}\frac{V_{(1,1)}(\tilde{\eta})-1}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}=\delta{\hat{\Delta}}_{r,s}^{(1,1)}(m)\frac{\log\left({x_{12}^{2}}/{a^{2}}\right)}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}+\frac{f^{(1,1)}_{r,s}(m)}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}~{},$
$\displaystyle\mathcal{I}^{\text{c.t.}}_{(1,1)}\equiv$
$\displaystyle-\frac{\alpha_{(1,1)}}{1-\Delta/2}\int_{-\infty}^{\infty}\frac{d{}x}{a}~{}\frac{V_{(1,1)}(\tilde{\eta})-1}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}\bigg{\rvert}_{y=a}=-\frac{2\delta{\hat{\Delta}}_{r,s}^{(1,1)}(m)}{1-\Delta/2}\frac{1}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}~{}.$
(4.18)

The function $f^{(1,1)}_{r,s}(m)$ does not seem to admit a simple closed-form
expression, but it is finite in the $m\rightarrow\infty$ limit.777One can
verify this by computing the integral directly at $m=\infty$, or by noticing
that the $O(a^{0})$ piece of the integrand reduces to a finite sum of $q=r-s$
terms at $m=\infty$. The coefficient of the logarithm in the first of eq.
(4.2.1) reads

$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(1,1)}(m)$
$\displaystyle=-\alpha_{(1,1)}\frac{\pi(m-1)(mr-ms+r-1)(mr-
ms+r+1)}{2^{1-\frac{4}{m+1}}(m-2)(m+1)(m+3)}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{k}\left(\frac{2}{m+1}\right)_{k}\left(\frac{2+r+rm-
sm+m}{m+1}\right)_{k}\left(\frac{2-r-rm+sm+m}{m+1}\right)_{k}}{k!(2)_{k}\left(\frac{3}{m+1}\right)_{k}\left(\frac{3}{2}+\frac{1}{m+1}\right)_{k}}~{}.$
(4.19)

For the $V_{(1,3)}$ contribution we have

$\displaystyle\alpha_{(1,3)}\int_{-\infty}^{\infty}d{}x\int_{a}^{\infty}\frac{d{}y}{y^{2}}~{}\frac{V_{(1,3)}(\tilde{\eta})}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}-\frac{\alpha_{(1,3)}}{1-\Delta/2}\int_{-\infty}^{\infty}\frac{d{}x}{a}~{}\frac{V_{(1,3)}(\tilde{\eta})}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}\bigg{\rvert}_{y=a}~{}.$
(4.20)

It is not difficult to show that, again up to terms that vanish as we send
$a\rightarrow 0$,

$\displaystyle\mathcal{I}_{(1,3)}$
$\displaystyle\equiv\alpha_{(1,3)}\int_{-\infty}^{\infty}d{}x\int_{a}^{\infty}\frac{d{}y}{y^{2}}~{}\frac{V_{(1,3)}(\tilde{\eta})}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}$
$\displaystyle=\frac{{f}^{(1,3)}_{\infty}(m)}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}\left(\frac{|x_{12}|}{a}\right)^{\frac{2}{1+m}}+\delta{\hat{\Delta}}_{r,s}^{(1,3)}(m)\frac{\log\left({x_{12}^{2}}/{a^{2}}\right)}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}+\frac{f^{(1,3)}_{r,s}(m)}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}~{},$
$\displaystyle\mathcal{I}^{\text{c.t.}}_{(1,3)}$
$\displaystyle\equiv-\frac{\alpha_{(1,3)}}{1-\Delta/2}\int_{-\infty}^{\infty}\frac{d{}x}{a}~{}\frac{V_{(1,3)}(\tilde{\eta})}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}\bigg{\rvert}_{y=a}$
$\displaystyle=-\frac{{f}^{(1,3)}_{\infty}(m)}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}\left(\frac{|x_{12}|}{a}\right)^{\frac{2}{1+m}}-\frac{2\delta{\hat{\Delta}}_{r,s}^{(1,3)}(m)}{1-\Delta/2}\frac{1}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}~{}.$
(4.21)

As is clear from the expression above, the power-law divergent terms cancel
out from the sum
$\mathcal{I}_{(1,3)}+\mathcal{I}_{(1,3)}^{\text{c.t.}}$.888The coefficient
that multiplies the power-law divergence in eq. (4.2.1) is
$\displaystyle{f}^{(1,3)}_{\infty}(m)=-\frac{\alpha_{(1,3)}}{\pi}\sin\left(\frac{\pi}{m+1}\right)\cos^{3}\left(\frac{\pi}{m+1}\right)\Gamma\left(-\frac{4}{m+1}\right)\Gamma\left(\frac{2}{m+1}\right)^{2}2^{\frac{3m+5}{m+1}}~{}.$
(4.22) The function $f^{(1,3)}_{r,s}(m)$, for which we do not have a simple
closed-form expression, behaves as

$\displaystyle f^{(1,3)}_{r,s}(m)$
$\displaystyle=-\frac{1}{2}\alpha_{(1,3)}m^{2}-{\alpha_{(1,3)}}{}\left(1+\log
2\right)m+O(m^{0})~{}.$ (4.23)

The coefficient of the logarithm in the first of eq. (4.2.1) reads

$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(1,3)}(m)$
$\displaystyle=\alpha_{(1,3)}\frac{\sqrt{\pi}\,\Gamma\left(-\frac{1}{m+1}\right)}{2^{1-\frac{2}{m+1}}\Gamma\left(\frac{1}{2}-\frac{1}{m+1}\right)}~{}$
$\displaystyle\quad\times\sum_{k=0}^{\infty}\frac{\left(-\frac{1}{m+1}\right)_{k}\left(\frac{1}{m+1}-\frac{1}{2}\right)_{k}\left(\frac{1}{2}+r-\frac{ms}{m+1}\right)_{k}\left(\frac{1}{2}-r+\frac{ms}{m+1}\right)_{k}}{k!\left(\frac{1}{2}-\frac{1}{m+1}\right)_{k}\left(\frac{3}{2}-\frac{1}{m+1}\right)_{k}\left(\frac{2}{m+1}-\frac{1}{2}\right)_{k}}~{}.$
(4.24)

The r.h.s. of eq. (4.2.1) features a ‘potential’ linear divergence at large
$m$ since $\Gamma\left(-\frac{1}{m+1}\right)\sim\Gamma(0)$ in this limit. On
general grounds, this divergence would be removed by a boundary counterterm of
the type $\sqrt{g}\int\psi_{(1,3)}$ with appropriately tuned coefficient,
which might induce an instability of the boundary condition, see e.g. [20]. In
our case this divergence is however compensated by the large-$m$ falloff of
$\alpha_{(1,3)}$ and, as we discussed previously, no counterterm is needed.

Finally, for the $V_{(1,5)}$ contribution we have

$\displaystyle\alpha_{(1,5)}\int_{-\infty}^{\infty}d{}x\int_{a}^{\infty}\frac{d{}y}{y^{2}}~{}\frac{V_{(1,5)}(\tilde{\eta})}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}-\frac{\alpha_{(1,5)}}{1-\Delta/2}\int_{-\infty}^{\infty}\frac{d{}x}{a}~{}\frac{V_{(1,5)}(\tilde{\eta})}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}\bigg{\rvert}_{y=a}~{},$
(4.25)

for which we find (up to regular terms as we send $a\rightarrow 0$ while
keeping $m$ finite)

$\displaystyle\mathcal{I}_{(1,5)}$
$\displaystyle\equiv\alpha_{(1,5)}\int_{-\infty}^{\infty}d{}x\int_{a}^{\infty}\frac{d{}y}{y^{2}}~{}\frac{V_{(1,5)}(\tilde{\eta})}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}~{}=\delta{\hat{\Delta}}_{r,s}^{(1,5)}(m)\frac{\log\left({x_{12}^{2}}/{a^{2}}\right)}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}+\frac{f^{(1,5)}_{r,s}(m)}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}~{},$
$\displaystyle\mathcal{I}^{\text{c.t.}}_{(1,5)}$
$\displaystyle\equiv-\frac{\alpha_{(1,5)}}{1-\Delta/2}\int_{-\infty}^{\infty}\frac{d{}x}{a}~{}\frac{V_{(1,5)}(\tilde{\eta})}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}{2^{\Delta}}}\bigg{\rvert}_{y=a}=-\frac{2\delta{\hat{\Delta}}_{r,s}^{(1,5)}(m)}{1-\Delta/2}\frac{1}{(x_{12}^{2})^{\hat{\Delta}_{r,s}}}~{}.$
(4.26)

The function $f^{(1,5)}_{r,s}(m)$ is constant in the $m\rightarrow\infty$
limit, and the coefficient of the logarithm in the first of eq. (4.2.1) reads

$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(1,5)}(m)$
$\displaystyle=\alpha_{(1,5)}\frac{\sqrt{\pi}\,\Gamma\left(\frac{3}{2}-\frac{3}{m+1}\right)}{2^{-\frac{2m}{m+1}}\Gamma\left(2-\frac{3}{m+1}\right)}\sum_{k=0}^{\infty}\frac{\left(\frac{m}{m+1}\right)_{k}\left(\frac{3}{2}-\frac{3}{m+1}\right)_{k}\left(\frac{2m+r+mr-
ms}{m+1}\right)_{k}\left(\frac{2m-r+ms-
mr}{m+1}\right)_{k}}{k!\left(\frac{3m}{m+1}\right)_{k}\left(2-\frac{3}{m+1}\right)_{k}\left(\frac{5}{2}-\frac{2}{m+1}\right)_{k}}~{}.$
(4.27)

Putting everything together, the one-loop, unrenormalized two-point
correlation function of $\psi_{(r,s)}$ at finite $m$ is

$\displaystyle
G_{1}(x_{12})=\mathcal{I}_{(1,1)}+\mathcal{I}^{\text{c.t.}}_{(1,1)}+\mathcal{I}_{(1,3)}+\mathcal{I}^{\text{c.t.}}_{(1,3)}+\mathcal{I}_{(1,5)}+\mathcal{I}^{\text{c.t.}}_{(1,5)}~{}.$
(4.28)

##### Renormalization and anomalous dimensions at finite $m$

Having obtained $G_{1}(x_{12})$, we can now renormalize it to extract the
anomalous dimension of $\psi_{(r,s)}$ at $O(g)$. At finite $m$, we can remove
the $\log\left({x_{12}^{2}}/{a^{2}}\right)$ divergence in eq. (4.28) via a
wave-function renormalization; i.e. we define $\psi^{R}_{(r,s)}\equiv
Z_{r,s}\psi_{(r,s)}$ with

$\displaystyle Z_{r,s}=1-\frac{1}{2}g\,\delta
Z_{r,s}\log\left({a^{2}/R^{2}}\right)+O(g^{2})~{},$ (4.29)

and choose

$\displaystyle\delta
Z_{r,s}=\delta{\hat{\Delta}}_{r,s}^{(1,1)}(m)+\delta{\hat{\Delta}}_{r,s}^{(1,3)}(m)+\delta{\hat{\Delta}}_{r,s}^{(1,5)}(m)~{}.$
(4.30)

The renormalized two-point correlation function at one-loop reads

$\displaystyle(x_{12}^{2})^{\hat{\Delta}_{r,s}}\langle\psi^{R}_{(r,s)}(x_{1})\psi^{R}_{(r,s)}(x_{2})\rangle=1-g\,\delta
Z_{r,s}\log\left({a^{2}/R^{2}}\right)-g\,(x_{12}^{2})^{\hat{\Delta}_{r,s}}G_{1}(x_{12})+O(g^{2})~{},$
(4.31)

and the one-loop anomalous dimension of $\psi_{r,s}$ is

$\displaystyle\delta\Delta_{r,s}=\delta Z_{r,s}~{}.$ (4.32)

##### The large-$m$ limit

The result of eq. (4.31) is valid as long as $m$ is kept finite. For $m$
large, there are additional divergences that should be removed from the
physical correlator before we can meaningfully take the large $m$ limit. As we
discuss in appendix B, at large $m$ and for $a_{1},a_{2}\ll m$ we have
that999On general grounds, $\alpha_{(1,3)}=O(m^{-1})$ follows from requiring
unphysical singularities in the correlator to cancel, see e.g. appendix B.5.1.
That $\alpha_{(1,5)}=O(m^{-2})$ follows from $B_{(1,3)}^{{\bf
a}\,(1,5)}=O(m^{-2})$ – see e.q. (B.3.1) – and upon assuming that the boundary
three-point functions ${\hat{C}}_{(r,s)(r,s)}^{{\bf a}(1,5)}$ remain finite in
the large-$m$ limit.

$\displaystyle\alpha_{(1,1)}=\sqrt{3}+O(m^{-1})~{},\quad\alpha_{(1,3)}={\alpha_{(1,3)}^{(1)}}{m^{-1}}+O(m^{-2}),\quad\alpha_{(1,5)}=O(m^{-2})~{},$
(4.33)

where

$\displaystyle\alpha_{(1,3)}^{(1)}=\frac{2\pi(r-s)(s-\text{sgn}(r-s))}{\sqrt{3}}~{}.$
(4.34)

From the explicit expressions given earlier in this section, it is not
difficult to verify that

$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(1,1)}(\infty)$
$\displaystyle=-\frac{\pi\sqrt{3}}{2}(r-s)^{2}\left(\frac{1}{3}+\frac{2}{3}|s-r|^{-1}\right)+O(m^{-1}),$
$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(1,3)}(\infty)$
$\displaystyle=-\frac{\alpha_{(1,3)}^{(1)}}{2}+O(m^{-1})~{},$
$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(1,5)}(\infty)$
$\displaystyle=O(m^{-2})~{}.$ (4.35)

With these results in hand we can easily verify that at large $m$ the
correlator of eq. (4.31) features a divergence linear in $m$. It originates
from the $V_{(1,1)}$ contribution to the counterterm and reads

$\displaystyle(x_{12}^{2})^{\delta{\hat{\Delta}}_{r,s}}(\mathcal{I}_{(1,1)}+\mathcal{I}^{\text{c.t.}}_{(1,1)})$
$\displaystyle=-m\,\delta h_{r,s}^{(1,1)}(\infty)+O(m^{0})~{}.$ (4.36)

We might expect further large-$m$ divergent contributions coming from
$\mathcal{I}_{(1,3)}+\mathcal{I}^{\text{c.t.}}_{(1,3)}$; however, thanks to a
special cancellation we find

$\displaystyle
f_{r,s}^{(1,3)}(m)-\frac{2\delta{\hat{\Delta}}_{r,s}^{(1,3)}(m)}{1-\Delta/2}=O(m^{0})~{}.$
(4.37)

We can remove the divergence of eq. (4.36) via an extra (multiplicative) wave-
function renormalization i.e. we define
$Z_{r,s}Z_{r,s}^{\text{extra}}\psi_{(r,s)}$ with $Z_{r,s}$ given by eq. (4.29)
and with

$\displaystyle Z_{r,s}^{\text{extra}}=1+\frac{1}{2}g\,m\,\delta
Z^{\text{extra}}_{r,s}+O(g^{2})~{},\quad\delta
Z^{\text{extra}}_{r,s}=-\delta{\hat{\Delta}}_{r,s}^{(1,1)}(\infty)~{},$ (4.38)

so that the renormalized one-loop two-point correlation function

$\displaystyle(x_{12}^{2})^{\hat{\Delta}_{r,s}}\langle\psi^{R}_{(r,s)}(x_{1})\psi^{R}_{(r,s)}(x_{2})\rangle=1$
$\displaystyle+gm\,\delta Z^{\text{extra}}_{r,s}-g\,\delta
Z_{r,s}\log\left({a^{2}/R^{2}}\right)$
$\displaystyle-g(x_{12}^{2})^{\hat{\Delta}_{r,s}}G_{1}(x_{12})+O(g^{2})~{},$
(4.39)

is completely finite. From the result above we can read off the boundary
spectrum at the IR fixed point:

$\displaystyle\hat{\Delta}^{\text{IR}}_{r,s}=\hat{\Delta}_{r,s}+g\delta{\hat{\Delta}}_{r,s}^{(1,1)}(\infty)+g\delta{\hat{\Delta}}_{r,s}^{(1,3)}(\infty)+O(m^{-2})=\frac{1}{4}(r-s)^{2}-\frac{r^{2}-s^{2}}{4m}+O(m^{-2})~{},$
(4.40)

where we used that

$\displaystyle\hat{\Delta}_{r,s}=\frac{1}{4}(r-s)^{2}+\frac{r^{2}-s^{2}}{4m}+O(m^{-2})~{}.$
(4.41)

##### Interpretation

The result of eq. (4.40) can be interpreted as follows. Assuming that the
endpoint of the RG flow is a conformal boundary condition (or a superposition
thereof) for the $\mathcal{M}_{m,m-1}$ minimal model, and assuming that
$\psi_{(r,s)}$ flows to another Virasoro primary
$\psi_{(r^{\prime},s^{\prime})}$ in such an IR conformal boundary condition,
then there should exist two positive integers $(r^{\prime},s^{\prime})$ such
that

$\displaystyle\hat{\Delta}_{r,s}-\hat{\Delta}_{r^{\prime},s^{\prime}}^{\text{IR}}=O(m^{-2}).$
(4.42)

By comparing equations (4.40) and (4.41), we find that if $r\neq s$ one
possible solution to the condition above is that
$(r^{\prime},s^{\prime})=(s,r)$. This would mean that

$\displaystyle\psi_{(r,s)}\longrightarrow\psi_{(s,r)}~{},$ (4.43)

under the RG flow.

This spectral flow is very natural: it would be consistent with the following
large-$m$ flow between conformal b.c.

$\displaystyle(a_{1},a_{2})_{m}\longrightarrow(a_{2},a_{1})_{m-1}~{},\quad
a_{1},a_{2}\ll m~{},$ (4.44)

since the selection rules imply that, if $(r,s)$ appears in
$(a_{1},a_{2})_{m}$ then $(s,r)$ appears in $(a_{2},a_{1})_{m-1}$. However we
should keep in mind that other possibilities (including flows to linear
combinations of conformal b.c.) are still logically possible. In particular,
it need not even be true that the infrared operator is still a Virasoro
primary.

##### Mixing

The reader may wonder why we do not consider operator mixing, since by
equation (4.41) there are often many operators whose dimensions become equal
at large $m$. In contrast with flat space there is however no need to consider
this mixing. This is because our one-loop result for the anomalous dimension
is reliable and physical (i.e. scheme-independent) for any finite $m$: after
all, the boundary is always conformal so $\hat{\Delta}$ is always a good
observable. At finite $m$ all the scaling dimensions are different, so the
matrix of two-point functions remains diagonal (cf. our discussion in
subsection 3.3) and there is no operator mixing. We have seen there are extra
divergences at large $m$ (those cancelled by the $\delta
Z^{\text{extra}}_{r,s}$ above) but obviously these can only appear if the
correlation function is non-zero in the first place. Therefore, in contrast
with a flat-space analysis where conformal invariance is completely lost along
the flow, nearby operators on the boundary of AdS do not mix.

###  4.3 Bulk one-point function of $\phi_{(1,3)}$

In this section we compute the one-point correlation function of
$\phi=\phi_{(1,3)}$ on the AdS disk

$\displaystyle
ds^{2}=\left(\frac{2R}{1-r^{2}}\right)^{2}(dr^{2}+r^{2}d\theta^{2})~{},\quad
0\leq r\leq 1~{},-\pi\leq\theta\leq\pi~{},$ (4.45)

at one loop. We again consider an elementary conformal boundary condition
$(a_{1},a_{2})_{m}$, with $a_{1},a_{2}\ll m$. At tree-level, putting for
simplicity $\phi$ at the center of the disk, we have (see eq. (B.3.1))

$\displaystyle(2R)^{\Delta}\langle\phi(0,0)\rangle_{0,{\bf
a}}={B_{(1,3)}^{{\bf a}\,(1,1)}}$
$\displaystyle=\left(\sqrt{3}-\frac{2\pi^{2}\left(2a_{2}^{2}-1\right)}{\sqrt{3}m^{2}}+\frac{4\pi^{2}\left(2a_{2}^{2}-1\right)}{\sqrt{3}m^{3}}+O(m^{-4})\right)~{},$
(4.46)

and the one-loop correction is given by

$\displaystyle\delta\langle\phi(0,0)\rangle_{{\bf a}}$
$\displaystyle=-g\mu^{2-\Delta}\int
d^{2}x\sqrt{g}\langle\phi(0,0)\phi(r,\theta)\rangle^{c}_{\bf a}$
$\displaystyle\qquad+\text{IR counterterms}+\text{UV counterterms}~{},$ (4.47)

where the superscript ‘c’ means that we will consider the connected correlator
only. Notice that the coupling $g$ is dimensionless and we inserted a scale
$\mu$.

##### Computing the integrand

As shown in appendix B.3, the (connected) correlator in the integrand of eq.
(4.3) reads

$\displaystyle(2R)^{2\Delta}\langle\phi(0,0)\phi(r,\theta)\rangle_{\bf
a}^{c}={\tilde{\mathcal{G}}(\tilde{\eta})-(B_{(1,3)}^{{\bf
a}\,(1,1)})^{2}}~{},$ (4.48)

with

$\displaystyle\tilde{\mathcal{G}}(\tilde{\eta})=(B_{(1,3)}^{{\bf
a}\,(1,1)})^{2}V_{(1,1)}(\tilde{\eta})+(B_{(1,3)}^{{\bf
a}\,(1,3)})^{2}V_{(1,3)}(\tilde{\eta})+(B_{(1,3)}^{{\bf
a}\,(1,5)})^{2}V_{(1,5)}(\tilde{\eta})~{},$ (4.49)

where

$\displaystyle V_{(1,1)}(\tilde{\eta})$
$\displaystyle=\,_{3}F_{2}\left(\frac{1-m}{m+1},\frac{2m}{m+1},-\frac{2m-2}{m+1};\frac{3+m}{2m+2},\frac{2-m}{m+1};-\frac{\tilde{\eta}}{4}\right)~{},$
$\displaystyle V_{(1,3)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{{h_{1,3}}/{2}}\,_{3}F_{2}\left(\frac{5m-1}{2m+2},\frac{1-m}{2m+2},-\frac{3m-3}{2m+2};\frac{3-m}{2m+2},\frac{3m+1}{2m+2};-\frac{\tilde{\eta}}{4}\right),$
$\displaystyle V_{(1,5)}(\tilde{\eta})$
$\displaystyle=\tilde{\eta}^{{h_{1,5}}/{2}}\,_{3}F_{2}\left(\frac{1}{m+1},\frac{m}{m+1},\frac{4m-1}{m+1};\frac{3m}{m+1},\frac{5m+1}{2m+2};-\frac{\tilde{\eta}}{4}\right)~{}.$
(4.50)

The cross-ratio $\tilde{\eta}$ reads

$\displaystyle\tilde{\eta}=\frac{\left(1-r^{2}\right)^{2}}{r^{2}}.$ (4.51)

We discuss separately the contributions from $V_{(1,1)}$, $V_{(1,3)}$, and
$V_{(1,5)}$.

##### The $V_{(1,1)}$ contribution

The relevant integrals in the first line of eq. (4.3) are

$\displaystyle
I_{(1,1)}\equiv\int_{\epsilon<r<\sqrt{1-a}}\sqrt{g}~{}(V_{(1,1)}(\tilde{\eta})-1)+c_{\text{c.t.}}\int\sqrt{\hat{g}}~{}(V_{(1,1)}(\tilde{\eta})-1)\big{\rvert}_{r=\sqrt{1-a}}~{},$
(4.52)

for a properly chosen $c_{\text{c.t.}}$, as explained below. For convenience
we have set an IR cut-off at $r=\sqrt{1-a}$ and a UV cut-off at $\epsilon>0$.
At finite $m$ the counterterm contribution is completely regular in the limit
of $a\rightarrow 0$:

$\displaystyle\int\sqrt{\hat{g}}~{}(V_{(1,1)}(\tilde{\eta})-1)\big{\rvert}_{r=\sqrt{1-a}}$
$\displaystyle=-\frac{4\pi
R\Delta^{2}(\Delta+2)}{(\Delta-4)(3\Delta-2)}a+O(a^{2})~{}.$ (4.53)

This is expected since there are no relevant boundary operators contributing
to $I_{(1,1)}$.

The bulk contribution is more subtle due to UV divergences. Upon invoking the
Mellin-Barnes representation of ${}_{3}F_{2}$:

$\,{}_{3}F_{2}(a_{1},a_{2},a_{3};b_{1},b_{2};x)=\frac{\Gamma(b_{1})\Gamma(b_{2})}{\Gamma(a_{1})\Gamma(a_{2})\Gamma(a_{3})}\int_{C}\frac{ds}{2\pi
i}\frac{(-x)^{-s}\Gamma(s)\Gamma(a_{1}-s)\Gamma(a_{2}-s)\Gamma(a_{3}-s)}{\Gamma(b_{1}-s)\Gamma(b_{2}-s)}~{},$
(4.54)

where the contour $C$ separates the poles $\Gamma(\dots+s)$ from those of
$\Gamma(\dots-s)$, we find

$\displaystyle\int_{\epsilon<r<\sqrt{1-a}}\sqrt{g}~{}{V_{(1,1)}(\tilde{\eta})}=\int_{C}\frac{ds}{2\pi
i}\,K_{(1,1)}(\Delta,s)\left[B_{1-a}(s+1,-2s-1)-B_{\epsilon^{2}}(s+1,-2s-1)\right]~{},$
(4.55)

where $B_{z}(a,b)$ is the incomplete beta function and

$\displaystyle
R^{-2}K_{(1,1)}(\Delta,s)\equiv\frac{4^{s}\pi\Delta\Gamma\left(\frac{1}{2}-\frac{3\Delta}{4}\right)\Gamma\left(-\frac{\Delta}{4}\right)\Gamma(s)\Gamma(-s-\Delta)\Gamma\left(-s-\frac{\Delta}{2}\right)\Gamma\left(1-s+\frac{\Delta}{2}\right)}{\Gamma\left(1-\frac{\Delta}{2}\right)\Gamma(-\Delta)\Gamma\left(\frac{\Delta}{2}\right)\Gamma\left(\frac{1}{2}-s-\frac{3\Delta}{4}\right)\Gamma\left(1-s-\frac{\Delta}{4}\right)}~{}.$
(4.56)

The contour $C$ is the black line shown in fig. 1 where the dots represent the
poles of the expression in eq. (4.55) (slightly displaced along the imaginary
axis for convenience of representation). As is clear from its definition in
terms of ${}_{2}F_{1}$, the $B_{z}(a,b)$ has poles at $-a=n\in\mathbb{N}$.

Figure 1:

For the $B_{1-a}(s+1,-2s-1)$ bit in (4.55), since

$\displaystyle B_{1-a}(s+1,-2s-1)=$
$\displaystyle\frac{1}{s+1}\left(\frac{a^{-2s-1}(1-a)^{s+1}\Gamma(s+2)\Gamma(2s+1)\,_{2}F_{1}(1,-s;-2s;a)}{\Gamma(s+1)\Gamma(2s+2)}+\frac{\Gamma(s+2)\Gamma(-2s-1)}{\Gamma(-s)}\right)~{},$
(4.57)

we shall close the contour to the left and pick up the residues for the poles
at $-s=n\in\mathbb{N}$, corresponding to the blue dots of fig. 1. The result
is

$\displaystyle\int_{C}\frac{ds}{2\pi
i}\,K_{(1,1)}(\Delta,s)B_{1-a}(s+1,-2s-1)=\frac{4\pi
R^{2}}{a}+R^{2}f_{\text{fin}}^{(1,1)}(m)+O(a).$ (4.58)

The finite term $f_{\text{fin}}^{(1,1)}(m)$ does not have a nice closed form,
but in the large-$m$ expansion it reads

$\displaystyle f_{\text{fin}}^{(1,1)}(m)=\frac{8}{3}\pi
m-8\pi-\frac{8\pi\left(15+2\pi^{2}\right)}{9m}+O(m^{-2})~{}.$ (4.59)

Note that the IR divergent term in the expression above drops out of the
connected correlator, since

$\displaystyle\int_{\epsilon<r<\sqrt{1-a}}\sqrt{g}=\frac{4\pi
R^{2}\left(a+\epsilon^{2}-1\right)}{a\left(\epsilon^{2}-1\right)}=\frac{4\pi
R^{2}}{a}-4\pi R^{2}+O(a,\epsilon)~{}.$ (4.60)

For the $B_{\epsilon^{2}}(s+1,-2s-1)$ bit in (4.55), since

$\displaystyle
B_{\epsilon^{2}}(s+1,-2s-1)=\frac{\epsilon^{2s+2}}{1+s}\left(1+\frac{2(s+1)^{2}\epsilon^{2}}{s+2}+O(\epsilon^{4})\right)~{},$
(4.61)

we shall close the contour to the right, and picking up residues at the red
poles in fig. 1 we find

$\displaystyle\int_{C}\frac{ds}{2\pi
i}\,K_{(1,1)}(\Delta,s)(-B_{\epsilon^{2}}(s+1,-2s-1))=\underset{\equiv
R^{2}f^{(1,1)}_{\epsilon}(m)}{\underbrace{\frac{4\pi
R^{2}\epsilon^{2-2\Delta}}{(\Delta-1)\left(1-2\cos\left(\frac{\pi\Delta}{2}\right)\right)}}}+O(\epsilon^{\\#})~{},$
(4.62)

where $O(\epsilon^{\\#})$ are positive powers of $\epsilon$, which will go to
zero as we send $\epsilon\rightarrow 0$ while keeping $\Delta$ finite. Putting
it all together, and taking the $a,\epsilon\rightarrow 0$ limit we get

$\displaystyle I_{(1,1)}$ $\displaystyle=4\pi
R^{2}+R^{2}f_{\text{fin}}^{(1,1)}(m)+R^{2}f^{(1,1)}_{\epsilon}(m)~{}.$ (4.63)

In particular the final result features a constant piece (in $m$) that comes
from taking the difference between (4.58) and (4.60).

##### The $V_{(1,3)}$ and the $V_{(1,5)}$ contributions

We proceed similarly for the other two contributions, but we skip some of the
details. For the $V_{(1,3)}$ contribution we have

$\displaystyle
I_{(1,3)}\equiv\int_{\epsilon<r<\sqrt{1-a}}\sqrt{g}~{}{V_{(1,3)}(\tilde{\eta})}+c_{\text{c.t.}}\int\sqrt{\hat{g}}~{}{V_{(1,3)}(\tilde{\eta})}\big{\rvert}_{r=\sqrt{1-a}}~{}.$
(4.64)

The counterterm contribution this time is divergent as $a\rightarrow 0$ and
equals

$\displaystyle\int\sqrt{\hat{g}}~{}{V_{(1,3)}(\tilde{\eta})}\big{\rvert}_{r=\sqrt{1-a}}=4\pi
Ra^{\frac{\Delta}{2}-1}(1+O(a))~{}.$ (4.65)

From the bulk term we find

$\displaystyle\int_{\epsilon<r<\sqrt{1-a}}\sqrt{g}~{}{V_{(1,3)}(\tilde{\eta})}$
$\displaystyle=\int_{C}\frac{ds}{2\pi
i}\,K_{(1,3)}(\Delta,s)\left(B_{1-a}\left(s-\frac{\Delta}{4}+1,-2s+\frac{\Delta}{2}-1\right)-B_{\epsilon^{2}}\left(s-\frac{\Delta}{4}+1,-2s+\frac{\Delta}{2}-1\right)\right)~{},$
(4.66)

where we defined

$\displaystyle
R^{-2}K_{(1,3)}(\Delta,s)\equiv-\frac{4^{s}\pi\Delta\Gamma\left(\frac{1}{2}-\frac{\Delta}{2}\right)\Gamma\left(\frac{\Delta}{4}\right)\Gamma(s)\Gamma\left(-s-\frac{3\Delta}{4}\right)\Gamma\left(-s-\frac{\Delta}{4}\right)\Gamma\left(1-s+\frac{3\Delta}{4}\right)}{\Gamma\left(1-\frac{3\Delta}{4}\right)\Gamma\left(-\frac{\Delta}{4}\right)\Gamma\left(\frac{3\Delta}{4}\right)\Gamma\left(\frac{1}{2}-s-\frac{\Delta}{2}\right)\Gamma\left(1-s+\frac{\Delta}{4}\right)}~{}.$
(4.67)

Besides the poles coming from $K_{(1,3)}(\Delta,s)$, in eq. (4.3) we have
additional poles from the incomplete beta functions at $s=\Delta/4-1$,
$\Delta/4-2,$ etc. The original contour $C$ has not moved, and we are free to
decide where these new poles lie with respect to $C$. A possible choice is
depicted in fig. 2.

Figure 2:

With this choice, for the
$B_{1-a}\left(s-\frac{\Delta}{4}+1,-2s+\frac{\Delta}{2}-1\right)$ bit we close
to the left to pick up residues at the blue and orange poles in fig. 2, while
for the
$B_{\epsilon^{2}}\left(s-\frac{\Delta}{4}+1,-2s+\frac{\Delta}{2}-1\right)$ we
close to the right, and all together we find

$\displaystyle\int_{\epsilon<r<\sqrt{1-a}}\sqrt{g}~{}$
$\displaystyle{V_{(1,3)}(\tilde{\eta})}{}=R^{2}f_{\text{fin}}^{(1,3)}(m)-\frac{8\pi
R^{2}a^{\frac{\Delta}{2}-1}}{\Delta-2}(1+O(a))+$
$\displaystyle\underset{\equiv
R^{2}f^{(1,3)}_{\epsilon}(m)}{\underbrace{\frac{2^{-\Delta}R^{2}\epsilon^{2-2\Delta}\sin\left(\frac{3\pi\Delta}{4}\right)\Gamma\left(\frac{3}{2}-\frac{\Delta}{2}\right)\Gamma\left(-\frac{3\Delta}{4}\right)\Gamma\left(\frac{\Delta}{4}\right)^{2}\Gamma\left(\frac{3\Delta}{2}+1\right)}{\sqrt{\pi}(\Delta-1)^{3}\Gamma(\Delta-1)\Gamma\left(-\frac{\Delta}{4}\right)}}}+O(\epsilon^{\\#})~{},$
(4.68)

where $O(\epsilon^{\\#})$ are positive powers of $\epsilon$, which will go to
zero as we send $\epsilon\rightarrow 0$ while keeping $\Delta$ finite and

$\displaystyle f_{\text{fin}}^{(1,3)}(m)=-\pi m-2\pi+O(m^{-1})~{}.$ (4.69)

It is easy to see that the counterterm contribution of eq. (4.65) cancels
exactly the IR divergence if we set

$\displaystyle c_{\text{c.t.}}=-\frac{R}{1-\frac{\Delta}{2}}~{}.$ (4.70)

Putting all together we find

$\displaystyle
I_{(1,3)}=R^{2}f^{(1,3)}_{\epsilon}(m)+R^{2}f_{\text{fin}}^{(1,3)}(m)~{}.$
(4.71)

Finally, we have for the $V_{(1,5)}$ contribution

$\displaystyle
I_{(1,5)}\equiv\int_{\epsilon<r<\sqrt{1-a}}\sqrt{g}~{}{V_{(1,5)}(\tilde{\eta})}{}+c_{c.t.}\int\sqrt{\hat{g}}~{}{V_{(1,5)}(\tilde{\eta})}{}\big{\rvert}_{r=\sqrt{1-a}}~{}.$
(4.72)

As expected, both the bulk term and counterterm are regular as $a\rightarrow
0$. From the bulk term we find, up to terms that vanish in the
$a,\epsilon\rightarrow 0$ limit

$\displaystyle\int_{\epsilon<r<\sqrt{1-a}}\sqrt{g}~{}{V_{(1,5)}(\tilde{\eta})}{}=$
$\displaystyle\underset{\equiv
R^{2}f^{(1,5)}_{\epsilon}(m)}{\underbrace{\frac{8R^{2}(\Delta+1)\epsilon^{2-2\Delta}\cos\left(\frac{\pi\Delta}{4}\right)\Gamma\left(\frac{1}{2}-\frac{\Delta}{4}\right)\Gamma\left(\frac{3\Delta}{4}+\frac{1}{2}\right)\Gamma\left(\frac{3}{4}(\Delta+2)\right)}{(\Delta-1)\Delta\Gamma\left(\frac{5\Delta}{4}+\frac{3}{2}\right)}}}+R^{2}f_{\text{fin}}^{(1,5)}(m)~{},$
(4.73)

with

$\displaystyle f_{\text{fin}}^{(1,5)}(m)=-4\pi m+\frac{8\pi}{3}+O(m^{-1})~{}.$
(4.74)

Hence

$\displaystyle
I_{(1,5)}=R^{2}f^{(1,5)}_{\epsilon}(m)+R^{2}f_{\text{fin}}^{(1,5)}(m)~{}.$
(4.75)

##### Cancellation of UV divergences

Putting together equations (4.63), (4.71) and (4.75), one finds

$\displaystyle\begin{split}(2R)^{\Delta}\delta\langle\phi(0,0)\rangle_{{\bf
a}}&=-g\mu^{2-\Delta}(2R)^{-\Delta}[(B_{(1,3)}^{{\bf
a}\,(1,1)})^{2}I_{(1,1)}+(B_{(1,3)}^{{\bf a}\,(1,3)})^{2}\
I_{(1,3)}+(B_{(1,3)}^{{\bf a}\,(1,5)})^{2}\ I_{(1,5)}]~{}\\\ &\qquad+\text{UV
counterterms}~{}.\end{split}$ (4.76)

To cancel the UV divergences we define the renormalized operator

$\displaystyle\phi^{R}=Z_{\phi}\phi+(2R)^{-\Delta}Z_{\phi{\mathds{1}}}{\mathds{1}}+\dots~{},$
(4.77)

and choose $Z_{\phi{\mathds{1}}}$ to cancel the power-law divergences, i.e.

$\displaystyle\begin{split}Z_{\phi{\mathds{1}}}&=\frac{1}{4}g(2\mu
R)^{2-\Delta}[(B_{(1,3)}^{{\bf
a}\,(1,1)})^{2}f^{(1,1)}_{\epsilon}(m)+(B_{(1,3)}^{{\bf
a}\,(1,3)})^{2}f^{(1,3)}_{\epsilon}(m)+(B_{(1,3)}^{{\bf
a}\,(1,5)})^{2}f^{(1,5)}_{\epsilon}(m)]\\\ &=g(2\mu
R)^{2-\Delta}\frac{\pi\epsilon^{2-2\Delta}}{\Delta-1}~{}.\end{split}$ (4.78)

Here the expression on the first line is what comes out of our computation,
which used the boundary conformal block decomposition. The expression on the
second line is what one would obtain from a short-distance expansion of the
bulk two-point function and isolating the divergent power. The fact that the
two expressions are identical is a consistency check for the bulk-boundary
crossing discussed in appendix B.3.1.

The wave-function renormalization factor $Z_{\phi}$ is obtained from a
standard bulk computation, and reads:

$\displaystyle Z_{\phi}=1+\delta Z_{\phi}$
$\displaystyle=1+\frac{1}{2}g\,C_{\phi\phi\phi}\frac{w_{\Delta}^{2}}{w_{2\Delta-2}}~{},\quad
w_{\Delta}=\frac{2^{2-\Delta}\pi\Gamma\left(\frac{2-\Delta}{2}\right)}{\Gamma(\Delta/2)}\,,$
$\displaystyle=1+g\,\left(\frac{2\pi
m}{\sqrt{3}}-\frac{4\pi}{\sqrt{3}}-\frac{10\pi}{\sqrt{3}m}+O(m^{-2})\right)~{},$
(4.79)

where in the last line we used $C_{\phi\phi\phi}$ of eq. (B.2). As we
discussed in subsection 3.1, in a general curved space one could have found
additional divergences (terms linear in $m$) due to mixing with
$(\text{Ric}){\mathds{1}}$ but their coefficient happens to vanish in AdS.

##### The final result

Collecting all the terms we ultimately find the following one-loop correction
to the one-point function of $\phi$:

$(2R)^{\Delta}\langle\phi(0,0)\rangle_{{\bf a}}=B_{(1,3)}^{{\bf
a}\,(1,1)}-2\pi g\left(m((2\mu
R)^{2-\Delta}-1)+\frac{1}{2}-\frac{8\pi^{2}(a_{2}^{2}+\ldots)}{3m}\right)+O(g^{2})~{}.$
(4.80)

We recognize the first term as due to the anomalous dimension of $\phi(x)$. It
can in principle be resummed to obtain an answer as a function of the running
coupling, but at the fixed point we may just set $(2\mu R)=1$ which removes
this term. Also at the fixed point we will have $g=O(m^{-1})$ so the
$O(m^{-1})$ term we wrote may in principle get corrected at two loops. For
this reason we did not write it out completely, although we did indicate the
dependence on the boundary label $a_{2}$ that follows from our computation.
Ultimately we obtain:

$(2R)^{\Delta}\langle\phi(0,0)\rangle_{{\bf a}}=B_{(1,3)}^{{\bf
a}\,(1,1)}-\frac{\sqrt{3}}{m}+\frac{16\pi^{2}a_{2}^{2}}{\sqrt{3}m^{2}}+O(m^{-2})~{},$
(4.81)

at the fixed point $g_{*}=\sqrt{3}/\pi m$. We recall that

$B_{(1,3)}^{{\bf
a}\,(1,1)}=\sqrt{3}-\frac{2\pi^{2}(2a_{2}^{2}-1)}{\sqrt{3}m^{2}}+O(m^{-3})\,.$
(4.82)

##### Interpretation

The interpretation of this result is perhaps a little disappointing. In flat
space, the $\phi_{(1,3)}$ operator flows to $\phi_{(3,1)}$ at the IR fixed-
point [13]. For the latter we have:

$(2R)^{\Delta}\langle\phi_{(3,1)}(0,0)\rangle_{(b_{1},b_{2})_{m-1}}=\sqrt{3}-\frac{2\pi^{2}(2b_{1}^{2}-1)}{\sqrt{3}m^{2}}+O(m^{-3})~{}.$
(4.83)

If $(a_{1},a_{2})_{m}$ flows to $(a_{2},a_{1})_{m-1}$ then we see that this
expression agrees with (4.82) to the given order, so we would naively expect
the one-loop correction to be zero. Equation (4.81) therefore presents us with
two issues: the non-zero correction multiplying $m^{-1}$ and the non-trivial
dependence on the boundary labels multiplying $m^{-2}$.

We believe that the first issue might be due to an incorrect identification of
the infrared operator. We have chosen the wave function renormalization factor
$Z_{\phi}$ such that the infrared operator is unit normalized in flat space,
so the overall normalization should be fine. However, as we mentioned above,
in curved space we can build new operators that involve the background
curvature. In particular we cannot exclude the possibility that the infrared
operator is any admixture of $\phi_{(3,1)}$ with the operator
$\text{Ric}{\mathds{1}}\sim R^{-2}{\mathds{1}}$. If this is the case then the
factor $\sqrt{3}/m$ in equation (4.81) merely amounts to a determination of
the relative coefficient between these two operators. We also note that such
an admixture should be visible at higher orders in perturbation theory,
because the latter operator has $\Delta=2$ exactly, whereas
$\Delta_{3,1}=2+4/m$.

The final term in (4.81) is subleading in $1/m$ and we could in principle have
dismissed it as unreliable. It was however argued in [15] that the dependence
on the boundary labels _is_ reliable at this order at least in flat space. If
this is so then it poses a puzzle. This term cannot be explained by operator
mixing, since the composition of the infrared operator should not depend on
the boundary labels. We also verified that the one-loop correction to
$\langle{\mathds{1}}\rangle$ does not feature any dependence on the boundary
labels at this order. However the analysis of [15] does not necessarily apply
in AdS, since it was based purely on the large-$m$ behavior of the various OPE
coefficients appearing in the computation. In our case we have seen that this
can be offset by the integrals themselves, which can diverge polynomially in
$m$. We therefore cannot conclude that this term is reliable without a more
detailed analysis of the integrals involved in the higher-loop contributions.

###  4.4 Irrelevant deformation

In this section we study the $\phi_{(3,1)}$ deformation of unitary and
diagonal minimal models $\mathcal{M}_{m+1,m}$ with conformal boundary
conditions ${\bf a}=(a_{1},a_{2})_{m}$, in $\text{AdS}_{2}$ and with $m>3$. We
will compute some anomalous dimensions of boundary Virasoro primaries at
finite $m$, at the first order in this deformation. In the large-$m$ limit,
with $a_{1},a_{2}\ll m$, the $\phi_{(3,1)}$ operator is the leading irrelevant
deformation, with scaling dimension

$\displaystyle\Delta=2h_{3,1}=2+\frac{4}{m}~{}.$ (4.84)

####  4.4.1 The perturbation

We consider the following bulk perturbation by $\phi\equiv\phi_{(3,1)}$

$\displaystyle\delta S=gR^{\Delta-2}\int
d^{2}x\sqrt{g}\,\phi(x+iy,x-iy)+\text{counterterms}+O(g^{2})~{}.$ (4.85)

Again, for large but finite $m$ we can remove the near-boundary divergences
without inducing a boundary RG flow, so that the only non-trivial $\beta$
function is for the bulk coupling and reads [31]

$\displaystyle\dot{g}=(2-\Delta)g-\pi\,C_{\phi\phi\phi}g^{2}+O(g^{3})~{}.$
(4.86)

At $m=\infty$ we have that $C_{\phi\phi\phi}={4}/{\sqrt{3}}$ (see eq. (B.6)),
and so using eq. (4.84) we find a UV fixed point at

$\displaystyle g_{*}=-\frac{\sqrt{3}}{\pi m}+O(m^{-2})~{}.$ (4.87)

In the bulk, this is the inverted RG flow between consecutive minimal models,
i.e. $\mathcal{M}_{m+1,m}\rightarrow\mathcal{M}_{m+2,m+1}$.

####  4.4.2 Anomalous dimensions

We present here some anomalous dimensions along the $\phi_{(3,1)}$ deformation
of unitary and diagonal minimal models, in $\text{AdS}_{2}$. The computations
for this case are entirely similar to those of section 4.2, so we omit the
details of the derivation.

The scaling dimension of $\psi_{(r,s)}$ after the perturbation of eq. (4.85)
(which goes towards the UV) reads

$\displaystyle\hat{\Delta}_{r,s}(g)=\hat{\Delta}_{r,s}+g\delta{\hat{\Delta}}_{r,s}^{(1,1)}(m)+g\delta{\hat{\Delta}}_{r,s}^{(3,1)}(m)+g\delta{\hat{\Delta}}_{r,s}^{(5,1)}(m)+O(g^{2})~{},$
(4.88)

where $\hat{\Delta}_{r,s}=h_{r,s}$ and

$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(1,1)}(m)$
$\displaystyle=B_{(3,1)}^{{\bf a}\,(1,1)}\frac{\pi(m+2)(mr-ms+r-1)(mr-
ms+r+1)}{2^{\frac{m+4}{m}}(2-m)m(m+3)}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{k}\left(-\frac{2}{m}\right)_{k}\left(\frac{r+m(r-s+1)-1}{m}\right)_{k}\left(\frac{-r+m(-r+s+1)-1}{m}\right)_{k}}{k!(2)_{k}\left(\frac{3}{2}-\frac{1}{m}\right)_{k}\left(-\frac{3}{m}\right)_{k}}~{},$
$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(3,1)}(m)$
$\displaystyle=\alpha_{(3,1)}\frac{\sqrt{\pi}\Gamma\left(\frac{1}{m}\right)}{2^{\frac{m+2}{m}}\Gamma\left(\frac{1}{2}+\frac{1}{m}\right)}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{\left(\frac{1}{m}\right)_{k}\left(-\frac{m+2}{2m}\right)_{k}\left(\frac{1}{2}+\frac{r}{m}+r-s\right)_{k}\left(\frac{1}{2}-\frac{(m+1)r}{m}+s\right)_{k}}{k!\left(\frac{1}{2}+\frac{1}{m}\right)_{k}\left(\frac{3}{2}+\frac{1}{m}\right)_{k}\left(-\frac{m+4}{2m}\right)_{k}}~{},$
$\displaystyle\delta{\hat{\Delta}}_{r,s}^{(5,1)}(m)$
$\displaystyle=\alpha_{(5,1)}\frac{\sqrt{\pi}2^{\frac{2}{m}+2}\Gamma\left(\frac{3}{2}+\frac{3}{m}\right)}{\Gamma\left(2+\frac{3}{m}\right)}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{\left(1+\frac{1}{m}\right)_{k}\left(\frac{3}{2}+\frac{3}{m}\right)_{k}\left(\frac{2+r+m(r-s+2)}{m}\right)_{k}\left(\frac{2-r+m(-r+s+2)}{m}\right)_{k}}{k!\left(\frac{5}{2}+\frac{2}{m}\right)_{k}\left(2+\frac{3}{m}\right)_{k}\left(3+\frac{3}{m}\right)_{k}}~{}.$
(4.89)

####  4.4.3 Large $m$

Finally we consider the large-$m$ limit with $a_{1},a_{2}\ll m$. In this limit
there are some extra $O(m)$ divergences, but they can be removed without
affecting the anomalous dimensions as we explained below eq. (4.36). The
large-$m$ expansion for the coefficients is given in equations (B.7) and
(B.78), recalling that $\alpha_{(5,1)}=O(m^{-2})$. Plugging in the value of
the UV fixed point in eq. (4.87) we find

$\displaystyle\hat{\Delta}^{\text{UV}}_{r,s}=\hat{\Delta}_{r,s}+g\delta{\hat{\Delta}}_{r,s}^{(1,1)}(\infty)+g\delta{\hat{\Delta}}_{r,s}^{(1,3)}(\infty)+O(m^{-2})=\frac{1}{4}(r-s)^{2}-\frac{r^{2}-s^{2}}{4m}+O(m^{-2})~{}.$
(4.90)

By comparing this result with the boundary spectrum of the
$\mathcal{M}_{m+2,m+1}$ minimal model we see that along the $\phi_{(3,1)}$
bulk RG flow (for $r\neq s$)

$\displaystyle\psi_{(r,s)}\longrightarrow\psi_{(s,r)}~{}.$ (4.91)

The spectral flow is again consistent with the following large-$m$ flow
between conformal b.c.:

$\displaystyle(a_{1},a_{2})_{m}\longrightarrow(a_{2},a_{1})_{m+1}~{},\quad
a_{1},a_{2}\ll m~{}.$ (4.92)

As expected, the $\phi_{(3,1)}$ deformation ‘inverts’ the RG flow of eq.
(4.44).

##  5 Outlook

We have analyzed RG flows in AdS both in general terms and in the explicit
example of the perturbed minimal models. To deal with the ultraviolet
divergences one employs familiar quantum field theory in curved space
techniques, and to deal with the infrared divergences one adds counterterms
familiar from holographic renormalization. In the end one obtains finite
renormalized correlation functions of both bulk and boundary operators.

In the future it would be interesting to resolve the somewhat puzzling status
of the one-point functions in the perturbed minimal models. This will be
essential for the identification of the correct infrared boundary conditions.
This would require computing $\langle\phi_{(1,3)}\rangle$ to two-loop order.
One might also compute $\langle\phi_{(r,s)}\rangle$ at one loop, for which the
level of difficulty should be similar to what was presented in this work. To
check our prescriptions at a structural level one might of course also try
other controllable RG flows like the Wilson-Fisher fixed points in AdS. (See
[5, 6] for some other examples. More generally there is a vast literature on
loop computations in AdS, but these mostly focus on Witten diagrams for
boundary correlation functions and not on bulk one- or higher-point
functions.)

This paper is not only an étude but also a prélude to the numerical study of
RG flows in AdS by applying conformal bootstrap methods to the boundary
correlation functions. The viability of this method was demonstrated in [1]
where initial results for scalar field theories in AdS2 were presented. In
upcoming work [2] we will analyze the minimal model flows in AdS numerically
and make essential use of the results presented above. In the future it will
be interesting to include the sum rules presented in [7] in the numerical
analysis.

## Acknowledgements

We would like to thank A. Antunes for many discussions and collaboration on
related topics and C. Behan, M. Billò, L. Di Pietro, A. Hebbar, M. Hogervorst,
M. Meineri, M. Reehorst, A. Tilloy and E. Trevisani for discussions. EL and
BvR are supported by the Simons Foundation grant $\\#$488659 (Simons
Collaboration on the non-perturbative bootstrap). This project is funded by
the European Union: for EL by ERC “QFT.zip” with Project ID 101040260 (held by
A. Tilloy) and for BvR by ERC “QFTinAdS” with Project ID 101087025. Views and
opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them.

##  Appendix A Two-dimensional BCFTs

We will consider two-dimensional CFTs on the upper half-plane

${\mathbb{H}^{+}}=\\{z\in\mathbb{C}\;|\;{\rm Im}~{}z\geq 0\\}.$

In Cartesian coordinates we write $z=x+iy$ so $x$ is also a boundary
coordinate.

We will consider various correlation functions between both bulk and boundary
global primaries in a generic BCFT on the upper half-plane.101010A Virasoro
primary or descendant operator is called a ‘global primary’ when it is a
primary with respect to the global conformal group $SL(2,\mathbb{C})$. In the
present subsection, $\phi_{i}(z,\bar{z})$ is a scalar bulk global primary with
dimension $\Delta_{i}$, and $\psi_{i}(x)$ is a scalar global boundary primary
with dimension $\hat{\Delta}_{i}$.

###  A.1 Global conformal invariance

We begin with the general structure that follows from $SL(2,\mathbb{C})$ bulk
invariance as well as the residual $SL(2,\mathbb{R})$ in the presence of a
boundary. The OPEs in the various channels are

$\displaystyle\phi_{i}(z_{1},\bar{z}_{1})\phi_{j}(z_{2},\bar{z}_{2})$
$\displaystyle=\sum_{k}C_{ij}{}^{k}\phi_{k}(z_{2},\bar{z}_{2})|z_{1}-z_{2}|^{\Delta_{k}-\Delta_{i}-\Delta_{j}}+\cdots\,,$
$\displaystyle\phi_{i}(x+iy,x-iy)$
$\displaystyle=\sum_{k}B_{i}{}^{k}\psi_{k}(x)(2y)^{\hat{\Delta}_{k}-\Delta_{i}}+\cdots\,,$
$\displaystyle\psi_{i}(x_{1})\psi_{j}(x_{2})$
$\displaystyle=\sum_{k}\hat{C}_{ij}{}^{k}\psi_{k}(x_{2})(x_{1}-x_{2})^{\hat{\Delta}_{k}-\hat{\Delta}_{i}-\hat{\Delta}_{j}}+\cdots\qquad(x_{1}>x_{2})\,.$
(A.1)

The ellipses in the first (second and third) lines above denote
$SL(2,\mathbb{C})$ ($SL(2,\mathbb{R})$) descendants. For both bulk and
boundary operators, indices are raised and lowered using the two-point
correlation function of the corresponding operator. Unless otherwise
specified, we will always consider unit-normalized primary bulk (boundary)
operators, and a unit-normalized identity operator:
$\langle{\mathds{1}}\rangle=\langle{\hat{\mathds{1}}}\rangle=1$. The values of
the OPE coefficients $B$ and $\hat{C}$ depend on the specific conformal
boundary condition.

We now provide the general form of some BCFT correlation functions as well as
the structure of their conformal block decomposition.

##### Simple correlation functions

On the upper half-plane the simplest non-trivial correlation functions are
($x_{ij}\equiv x_{i}-x_{j}$ and
$\hat{\Delta}_{ijk}\equiv\hat{\Delta}_{i}+\hat{\Delta}_{j}-\hat{\Delta}_{k}$)

$\displaystyle\langle\psi(x_{1})\psi(x_{2})\rangle_{{\mathbb{H}^{+}}}$
$\displaystyle=\frac{1}{(x_{12}^{2})^{\hat{\Delta}}}~{},\qquad\qquad\qquad\qquad\quad(x_{1}>x_{2})~{}$
$\displaystyle\langle\phi(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}$
$\displaystyle=\frac{B_{\phi}}{(2y)^{\Delta}}~{},\qquad\qquad\qquad\qquad\quad\quad(y>0)~{}$
$\displaystyle\langle\phi(x_{1}+iy_{1},x_{1}-iy_{1})\psi(x_{2})\rangle_{{\mathbb{H}^{+}}}$
$\displaystyle=\frac{B_{\phi\psi}}{(2y_{1})^{\Delta-\hat{\Delta}}(x_{12}^{2}+y_{1}^{2})^{\hat{\Delta}}}~{},~{}\qquad\quad(y_{1}>0)~{}$
$\displaystyle\langle\psi_{i}(x_{1})\psi_{j}(x_{2})\psi_{k}(x_{3})\rangle_{{\mathbb{H}^{+}}}$
$\displaystyle=\frac{\hat{C}_{ijk}}{(x_{12})^{\hat{\Delta}_{ijk}}(x_{23})^{\hat{\Delta}_{jki}}(x_{13})^{\hat{\Delta}_{ikj}}}~{},\quad(x_{i}>x_{i+1})~{}.$
(A.2)

##### Boundary four-point function

Consider the four-point correlation function between four global boundary
conformal primaries $\psi_{i}$ with scaling dimensions $\hat{\Delta}_{i}$. By
$SL(2,\mathbb{R})$ symmetry this correlation function takes the form (here and
in the following $\hat{\Delta}_{ij}\equiv\hat{\Delta}_{i}-\hat{\Delta}_{j}$)

$\displaystyle\langle\psi_{1}(x_{1})\psi_{2}(x_{2})\psi_{3}(x_{3})\psi_{4}(x_{4})\rangle_{{\mathbb{H}^{+}}}=\left(\frac{x_{14}}{x_{24}}\right)^{\hat{\Delta}_{21}}\left(\frac{x_{14}}{x_{13}}\right)^{\hat{\Delta}_{34}}\frac{\mathcal{G}^{1234}(\eta)}{(x_{12})^{\hat{\Delta}_{1}+\hat{\Delta}_{2}}(x_{34})^{\hat{\Delta}_{3}+\hat{\Delta}_{4}}}~{},\quad
x_{i}>x_{i+1}~{},$ (A.3)

with cross-ratio

$\displaystyle\eta=\frac{x_{12}x_{34}}{x_{13}x_{24}}~{},\quad 0<\eta<1~{}.$
(A.4)

In the s-channel we have

$\displaystyle\mathcal{G}^{1234}(\eta)=\sum_{k}\hat{C}_{12}{}^{k}\hat{C}_{34k}G(\hat{\Delta}_{21},\hat{\Delta}_{34},\hat{\Delta}_{k},\eta)~{},$
(A.5)

with global blocks given by [32]

$\displaystyle
G(a,b,\Delta,\eta)=\eta^{\Delta}\,_{2}F_{1}(a+\Delta,b+\Delta;2\Delta;\eta)~{}.$
(A.6)

##### Bulk two-point function

Consider the two-point correlation function between two bulk scalar global
primaries $\phi_{i}$ with scaling dimensions $\Delta_{i}$. By
$SL(2,\mathbb{R})$ symmetry this two-point correlation function takes the
following form [33, 34, 18]:

$\displaystyle\langle\phi_{1}(x_{1}+iy_{1},x_{1}-iy_{1})\phi_{2}(x_{2}+iy_{2},x_{2}-iy_{2})\rangle_{{\mathbb{H}^{+}}}=\frac{f(\xi)}{(2y_{1})^{\Delta_{1}}(2y_{2})^{\Delta_{2}}}~{}.$
(A.7)

The cross-ratio is

$\displaystyle\xi\equiv\frac{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}{4y_{1}y_{2}}~{}.$
(A.8)

The bulk and boundary channel expansions of $f(\xi)$ read

$\displaystyle
f(\xi)=\xi^{-\frac{1}{2}(\Delta_{1}+\Delta_{2})}\sum_{k}C_{12}{}^{k}B_{k}f_{\text{bulk}}(\Delta_{k};\xi)=\sum_{k}B_{1}{}^{k}B_{2k}~{}f_{\text{bdy}}(\hat{\Delta}_{k};\xi)~{},$
(A.9)

where [33, 34, 18]

$\displaystyle f_{\text{bulk}}(\Delta_{k};\xi)$
$\displaystyle=\xi^{\Delta_{k}/2}{}F_{1}\left(\frac{\Delta_{k}}{2},\frac{\Delta_{k}}{2},\Delta_{k};-\xi\right)~{},$
$\displaystyle f_{\text{bdy}}(\hat{\Delta}_{k};\xi)$
$\displaystyle=\xi^{-\hat{\Delta}_{k}}{}F_{1}\left(\hat{\Delta}_{k},\hat{\Delta}_{k},2\hat{\Delta}_{k};-{1}/{\xi}\right)~{}.$
(A.10)

##### Bulk-boundary-boundary function

Consider the correlation function between two boundary global primaries of
scaling dimension $\hat{\Delta}_{1,2}$ and one (scalar) bulk global primary of
scaling dimension $\Delta$. Via the bulk-boundary OPE of $\phi$, this
correlator can be written as

$\displaystyle\langle\psi_{1}({x}_{1})\psi_{2}({x}_{2})$
$\displaystyle\phi(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=$
$\displaystyle\frac{\sum_{k}B_{\phi}{}^{k}{\hat{C}}_{12k}f(\hat{\Delta}_{12},\hat{\Delta}_{k},v)}{(y^{2}+(x_{1}-x)^{2})^{{\hat{\Delta}_{12}/2}}(y^{2}+(x_{2}-x)^{2})^{{\hat{\Delta}_{21}/2}}|{x}_{12}|^{\hat{\Delta}_{1}+\hat{\Delta}_{2}}(2y)^{\Delta}}~{}.$
(A.11)

In our conventions the blocks computed in refs. [28, 29] read

$\displaystyle
f(\hat{\Delta}_{12},\hat{\Delta}_{k},v)=(4v)^{\hat{\Delta}_{k}/2}\,_{2}F_{1}\left(\frac{\hat{\Delta}_{k}+\hat{\Delta}_{12}}{2},\frac{\hat{\Delta}_{k}-\hat{\Delta}_{12}}{2};\hat{\Delta}_{k}+\frac{1}{2};v\right)~{},$
(A.12)

with

$\displaystyle
v=\frac{y^{2}({x}_{12})^{2}}{(y^{2}+({x}_{1}-{x})^{2})(y^{2}+({x}_{2}-{x})^{2})}\equiv\tilde{\eta}/4~{},\quad
0\leq v\leq 1~{}.$ (A.13)

###  A.2 Local conformal invariance

Let $T_{\mu\nu}$ be the bulk stress tensor. Conservation and tracelessness
imply

$\displaystyle\bar{\partial}T=0~{},\quad{\partial}\bar{T}=0~{},$ (A.14)

where $\partial\equiv\partial_{z}$, $\bar{\partial}\equiv\partial_{\bar{z}}$
and we have defined

$\displaystyle T\equiv T(z)\equiv
T_{zz}(z)~{},\quad\bar{T}\equiv\bar{T}({\bar{z}})\equiv
T_{{\bar{z}}\bar{z}}(\bar{z}).$ (A.15)

Translation invariance along the boundary is equivalent to Cardy’s condition
[35, 36, 37]

$\displaystyle T(z)=\bar{T}({\bar{z}})~{},\quad{\rm Im}~{}z=0~{}.$ (A.16)

It follows from equations (A.14) and (A.16) that the boundary limit of $T$ is
a global boundary primary operator of scaling dimension $\hat{\Delta}=2$, i.e.

$\displaystyle\text{D}(x)=T(x+iy)\rvert_{y=0}~{}.$ (A.17)

We call this boundary operator the ‘displacement’. The restriction of the
anti-holomorphic component $\bar{T}$ of the stress tensor to the boundary
defines the same displacement, thanks to Cardy’s condition.

####  A.2.1 Conformal Ward identities on the upper half-plane

Let us consider an infinitesimal real-analytic transformation on the upper
half-plane ${\mathbb{H}^{+}}$

$\displaystyle z\rightarrow
z+\epsilon(z),~{}\quad\epsilon(z)=\sum_{n}a_{n}z^{n+1}~{},\quad
a_{n}\in\mathbb{R}~{}.$ (A.18)

The conserved charge on ${\mathbb{H}^{+}}$ is

$\displaystyle Q_{\epsilon}=\frac{1}{2\pi
i}\oint_{C^{+}}\epsilon(z)T(z)dz-\frac{1}{2\pi
i}\oint_{C^{+}}\epsilon(\bar{z})\bar{T}(\bar{z})d\bar{z}~{},$ (A.19)

where $C^{+}$ is a contour in ${\mathbb{H}^{+}}$, we regarded
$\epsilon(\bar{z})$ as $\epsilon^{*}(z)$ and $\bar{T}(\bar{z})$ is the
analytic continuation of $T(z)$ to the lower half-plane (${\mathbb{H}^{-}}$).
Because of Cardy’s condition (A.16), the operator $Q_{\epsilon}$ is well
defined even when correlation functions are evaluated on the real axis. We can
then define the Virasoro generators in analogy with those in the full complex
plane as

$\displaystyle L_{n}$ $\displaystyle=\frac{1}{2\pi
i}\oint_{C^{+}}z^{n+1}T(z)dz-\frac{1}{2\pi
i}\oint_{C^{+}}{\bar{z}}^{n+1}\bar{T}(\bar{z})d\bar{z}~{}$
$\displaystyle=\frac{1}{2\pi i}\oint_{C\in\mathbb{C}}z^{n+1}T(z)dz~{},$ (A.20)

where in the last line we used Cardy’s condition (A.16) to combine the two
addends into a single contour integral on the full complex plane. The
generators so defined then obey the commutation relations of a holomorphic
Virasoro algebra, with central charge $c$

$\displaystyle[L_{n}\,,L_{m}]=(n-m)L_{n+m}+\frac{c}{12}n(n^{2}-1)\delta_{n+m,0}~{}.$
(A.21)

The elements $\\{L_{\pm 1},L_{0}\\}$ of the Virasoro algebra generate the
conformal group $SL(2,\mathbb{R})$. On the real axis, we can define Virasoro
primary operators $\psi_{i}$ with respect to this holomorphic Virasoro algebra
as

$[L_{0},\psi_{i}(0)]=\hat{\Delta}_{i}\,\psi_{i}(0)~{},\quad[L_{n},\psi_{i}(0)]=0~{},\quad
n\geq 1~{}.$ (A.22)

Using the definition above it follows that, under the (finite) real-analytic
conformal transformations

$\displaystyle z\rightarrow
f(z)~{},\quad\psi_{i}^{\prime}(f(z))=\left(\frac{\partial f}{\partial
z}\right)^{-\hat{\Delta}_{i}}\psi_{i}(z)~{}.$ (A.23)

Namely $\psi_{i}$ transforms as a ‘holomorphic Virasoro primary’, by which we
mean a Virasoro primary with weights
$h_{i}=\hat{\Delta}_{i},~{}\bar{h}_{i}=0$. Equivalently eq. (A.2.1) implies
that

$\displaystyle
T(z)\psi_{i}(w)=\frac{\hat{\Delta}_{i}}{(z-w)^{2}}{\psi_{i}}(w)+\frac{1}{z-w}\partial{\psi_{i}}(w)+O((z-w)^{0})~{}.$
(A.24)

Boundary Virasoro descendants are obtained by repeatedly applying the
$L_{-k}$’s with $k>0$ to the primary ${\psi_{i}}$. Using eq. (A.2.1) we have
that

$\displaystyle[L_{-k},{\psi_{i}}(w)]\equiv{\psi_{i}}^{(-k)}(w)=\frac{1}{2\pi
i}\oint_{C_{w}}\,dz\,(z-w)^{1-k}T(z){\psi_{i}}(w)~{},$ (A.25)

where the contour $C_{w}$ lies on the full complex plane and encircles $w$. We
can use eq. (A.25) to compute the correlation function of
${\psi_{i}}^{(-k)}(w)$ with any number of either bulk or boundary Virasoro
primaries from the correlation function of ${\psi_{i}}(w)$ with the same
number of either bulk or boundary Virasoro primaries. In terms of the string
of Virasoro primaries (in defining ${\mathcal{X}}$ we disregard its anti-
holomorphic dependence as it plays no role in the following)

$\displaystyle{\mathcal{X}}^{(n,m)}(\\{z_{i}\\})\equiv\psi_{1}(z_{1})\dots\psi_{n}(z_{n})\phi_{n+1}(z_{n+1},\bar{z}_{n+1})\dots\phi_{n+m}(z_{n+m},\bar{z}_{n+m})~{},$
(A.26)

we have that

$\displaystyle\langle\psi^{(-k)}(w)$
$\displaystyle{\mathcal{X}}^{(n,m)}(\\{z_{i}\\})\rangle$
$\displaystyle=\frac{1}{2\pi i}\oint_{C_{w}}\frac{dz}{(z-w)^{k-1}}\langle
T(z)\psi(w){\mathcal{X}}^{(n,m)}(\\{z_{i}\\})\rangle~{},$
$\displaystyle=-\frac{1}{2\pi
i}\oint_{C_{\\{z_{i}\\}}}\frac{dz}{(z-w)^{k-1}}\langle
T(z)\psi(w){\mathcal{X}}^{(n,m)}(\\{z_{i}\\})\rangle~{},$
$\displaystyle=-\frac{1}{2\pi
i}\oint_{C_{\\{z_{i}\\}}}\frac{dz}{(z-w)^{k-1}}\sum_{i=1}^{n+m}\left(\frac{1}{z-z_{i}}\partial_{i}+\frac{h_{i}}{(z-z_{i})^{2}}\right)\langle\psi(w){\mathcal{X}}^{(n,m)}(\\{z_{i}\\})\rangle~{},$
$\displaystyle=\mathcal{L}_{-k}^{(w)}~{}\langle\psi(w){\mathcal{X}}^{(n,m)}(\\{z_{i}\\})\rangle~{},$
(A.27)

and we denoted the weight of each boundary Virasoro primary $\psi_{i}$ as
$\hat{\Delta}_{i}=h_{i}$. In the third line we modified the integration
contour in order to pick up the singularities at $z=z_{i}$. These
singularities are simple and double poles, as dictated by the OPE of $T$ with
either $\phi_{i}$ or $\psi_{i}$

$\displaystyle T(z){\mathcal{X}}^{(n,m)}(\\{z_{i}\\})\underset{z\rightarrow
z_{k}}{\sim}\frac{h_{i}}{(z-z_{k})^{2}}{\mathcal{X}}^{(n,m)}(\\{z_{i}\\})+\frac{1}{z-z_{k}}\partial_{k}{\mathcal{X}}^{(n,m)}(\\{z_{i}\\})+\text{regular}~{}.$
(A.28)

In the last step, after computing the integral using the residue theorem, we
introduced the important differential operator

$\displaystyle\mathcal{L}_{-k}^{(w)}\equiv\sum_{i=1}^{n+m}\left(\frac{(k-1)h_{i}}{(z_{i}-w)^{k}}-\frac{1}{(z_{i}-w)^{k-1}}\partial_{i}\right)~{}.$
(A.29)

The result in eq. (A.2.1) implies that if $\psi^{(-k)}$ is a null descendant
of $\psi$ at level $k$, correlation functions with $\psi$ insertions satisfy a
linear differential equation of order $k$. By construction, this will be the
same differential equation satisfied by a holomorphic Virasoro primary with
weight $(h=\hat{\Delta},~{}\bar{h}=0)$.

####  A.2.2 The method of images

Bulk correlation functions on the upper half-plane ${\mathbb{H}^{+}}$ can be
computed using the method of images [35] (see also Chapter 11.2 of [38]) and
bulk-boundary crossing symmetry. To see how it works, let us consider the
effect of the transformation (A.18) on a correlator between a string of bulk
Virasoro primaries

$\displaystyle{\mathcal{X}}(z_{1},\bar{z}_{1};\dots;z_{n},\bar{z}_{n})\equiv\phi_{1}(z_{1},\bar{z}_{1})\dots\phi_{n}(z_{n},\bar{z}_{n})~{},$
(A.30)

with $\Delta_{i}=h_{i}+\bar{h}_{i}$. Note that here $h_{i}\neq\bar{h}_{i}$,
generically. This transformation is (recall eq. (A.19))

$\displaystyle\frac{1}{2\pi i}\oint_{C^{+}}\epsilon(z)dz\langle
T(z){\mathcal{X}}(z_{1},\bar{z}_{1};\dots;z_{n},\bar{z}_{n})\rangle_{{\mathbb{H}^{+}}}-\frac{1}{2\pi
i}\oint_{C^{+}}\epsilon(\bar{z})d\bar{z}\langle\bar{T}(\bar{z}){\mathcal{X}}(z_{1},\bar{z}_{1};\dots;z_{n},\bar{z}_{n})\rangle_{{\mathbb{H}^{+}}}~{}.$
(A.31)

We can regard the dependence of the correlators on antiholomorphic coordinates
$\bar{z}_{i}$ on the upper half-plane ${\mathbb{H}^{+}}$ as a dependence on
the holomorphic coordinates $z_{i}^{*}=\bar{z}_{i}$ on the lower half-plane
${\mathbb{H}^{-}}$. This ‘mirroring’ is nothing but a parity transformation
with respect to the real axis, and as such holomorphic indices change into
antiholomorphic ones e.g. $T(z^{*})=\bar{T}(z)$.111111If a given bulk operator
is purely holomorphic, so $\bar{h}=0$, then its antiholomorphic part is the
identity operator. As a consequence there is no need to double holomorphic
operators, such as $T(z)$. Since $T=\bar{T}$ on the real axis by Cardy’s
condition, when we combine the two addends above to write a contour integral
over the full complex plane, the total contribution from the real axis
vanishes, and we are left with a contour integral of a correlator with $2n$
holomorphic Virasoro primaries on the full complex plane:

$\displaystyle\frac{1}{2\pi i}\oint_{C}\epsilon(z)dz\langle
T(z){\mathcal{X}}(z_{1},z^{\prime}_{1};\dots;z_{n},z^{\prime}_{n})\rangle~{},\quad
z_{i}^{\prime}\equiv z^{*}_{i}~{}.$ (A.32)

Hence, the correlation function
$\langle\phi_{1}(z_{1},\bar{z}_{1})\dots\phi_{n}(z_{n},\bar{z}_{n})\rangle_{{\mathbb{H}^{+}}}$
regarded as a function of $(z_{1},\bar{z}_{1};\dots;z_{n},\bar{z}_{n})$
satisfies the same differential equation as the correlation function on
homogeneous space,
$\langle\phi_{1}(z_{1},\bar{z}_{1})\dots\phi_{n}(z_{2n},\bar{z}_{2n})\rangle$,
as a function of $(z_{1},\dots,z_{2n})$.

As an application, let us compute the one-point correlation function on
${\mathbb{H}^{+}}$ of a Virasoro primary $\phi$ of weight $\Delta=h+\bar{h}$
and spin $\ell=h-\bar{h}$, denoted as $\phi_{h,\bar{h}}(z,\bar{z})$.121212The
attentive reader will have noticed a small change of notation: here and in the
next section the Virasoro primaries are labelled by their dimensions
$(h,\bar{h})$, and not by a pair of integers $(r,s)$ as everywhere else in
this work. This choice is convenient in order to explain the application of
the method of images. By the method of images, we are instructed to consider

$\displaystyle\langle\phi_{h,\bar{h}}(z,\bar{z})\rangle_{{\mathbb{H}^{+}}}\propto\langle\phi_{h}(z)\phi_{\bar{h}}(z^{*})\rangle~{},$
(A.33)

where we have regarded $z^{*}$ as a holomorphic coordinate, and
$\phi_{\bar{h}}$ as a holomorphic field with holomorphic dimension $\bar{h}$ .
Note that the $\propto$ symbol is appropriate, since the method of images
cannot fix overall constants in correlation functions. Now, the two-point
correlation function on the right-hand side above is fixed by conformal
symmetry as in eq. (A.42). In particular we immediately learn that if
$\ell\neq 0$, then $\phi$ has a vanishing one-point correlation function on
${\mathbb{H}^{+}}$. Alternatively, if $\ell=0$, writing $z=x+iy$ with $y>0$,
we get

$\displaystyle\langle\phi_{h,{h}}(x+iy,x-iy)\rangle_{{\mathbb{H}^{+}}}=\frac{B_{\phi}}{(2y)^{\Delta}}~{},\quad\Delta=2h~{}.$
(A.34)

The constant $B_{\phi}\equiv B_{\phi{\mathds{1}}}$, which is not fixed by the
method of images, is a physical datum of the BCFT. Finally let us consider the
bulk-boundary two-point correlation function between the scalar bulk Virasoro
primary $\phi$ and a boundary Virasoro primary $\psi$ with scaling dimension
$\hat{\Delta}$. Applying again the method of images, we compute

$\displaystyle\langle\phi_{h,{h}}(z,\bar{z})\psi(0)\rangle_{{\mathbb{H}^{+}}}\propto\langle\phi_{h}(z)\phi_{h}(z^{*})\psi(0)\rangle~{},$
(A.35)

where we have regarded $\psi$ as a holomorphic operator with holomorphic
dimension $\hat{\Delta}$. Choosing again $z=x+iy$ with $y>0$, we then find

$\displaystyle\langle\phi_{h,{h}}(x+iy,x-iy)\psi(0)\rangle_{{\mathbb{H}^{+}}}=\frac{B_{\phi\psi}}{(2y)^{\Delta-\hat{\Delta}}(y^{2}+x^{2})^{\hat{\Delta}}}~{}.$
(A.36)

The constant $B_{\phi\psi}$ is another physical datum of the BCFT.

###  A.3 Some universal correlation functions

Correlation functions involving the stress tensor $T(z)$, $\bar{T}(\bar{z})$
and the displacement operator $\text{D}(x)$ are to a large extent fixed by the
conformal Ward identities. In the instances below we will find several such
correlation functions. They are universal in the sense that they do not depend
on the specific BCFT and instead only depend on $c$ and the scaling dimension
of the operators involved. By taking coincident points we also obtain
correlation functions of $\text{D}^{2}(x)$ and $T\bar{T}(z,\bar{z})$.

####  A.3.1 Interlude: the stress tensor on the plane

In a generic 2d CFT, correlation functions with stress tensor insertions are
determined by the conformal Ward identities [39]. Consider first the simple
case with just one $T$ insertion in between a string of $n$ Virasoro primaries
$\phi_{i}(z_{i},\bar{z}_{i})$, with weights $(h_{i},\bar{h}_{i})$. In the
notation introduced in eq. (A.26), this correlator will be denoted as

$\displaystyle\langle T(w){\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle~{}.$
(A.37)

Because the $T$ is a level-two descendant of the identity operator
${\mathds{1}}$, from eq. (A.2.1) we immediately find

$\displaystyle\langle
T(w){\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle=\mathcal{L}_{-2}^{(w)}\langle{\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle~{}.$
(A.38)

Let us now consider two insertions of the stress tensor.

$\displaystyle\langle
T(w)T(w^{\prime}){\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle~{}.$ (A.39)

We can apply the same procedure that led to the formula in eq. (A.2.1) to
reduce the number of $T$ insertions, but this time we get additional
contributions from the additional poles in the $T\times T$ OPE when deforming
the integration contour. We recall that

$\displaystyle
T(z)T(0)=\frac{c/2}{z^{4}}{\mathds{1}}+\frac{2}{z^{2}}T(0)+\frac{1}{z}T^{\prime}(0)+\frac{3}{10}T^{\prime\prime}(0)+T^{2}(0)+O(z^{2})~{},$
(A.40)

where primed operators are derivatives with respect to $z$, $O(z^{2})$ denotes
higher order contributions and $T^{2}$ is the unique global primary at level
four in the $T\times T$ OPE.131313This operator is obtained (up to
normalization) by acting with $L_{-4}-\frac{5}{3}L_{-2}^{2}$ on the Virasoro
identity module. Using the $T\times T$ OPE we then get

$\displaystyle\langle T(w)T(w^{\prime})$
$\displaystyle{\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle=\mathcal{L}_{-2}^{(w)}\langle
T(w^{\prime}){\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle+\frac{c/2}{(w-w^{\prime})^{4}}\langle{\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle~{},$
(A.41)

and $\mathcal{L}_{-2}^{(w)}$ acts on $T(w^{\prime})$, as well as on the string
${\mathcal{X}}$. As a particular case of the equations above, let us take
${\mathcal{X}}$ to be a string of two identical holomorphic Virasoro primaries
with two-point function

$\displaystyle\langle\phi_{i}(z_{1}){\phi_{i}}(z_{2})\rangle=\frac{1}{(z_{12})^{2h_{i}}}~{},\quad
z_{ij}\equiv z_{i}-z_{j}~{}.$ (A.42)

From eq. (A.38) we obtain

$\displaystyle\langle
T(z_{1})\phi_{i}(z_{2}){\phi_{i}}(z_{3})\rangle=\frac{C_{T\phi\phi}}{(z_{12})^{2}(z_{23})^{2h_{i}-2}(z_{31})^{2}}~{},\quad
C_{T\phi\phi}=h_{i}~{}.$ (A.43)

From eq. (A.41) we obtain

$\displaystyle\langle$ $\displaystyle
T(z_{0})T(z_{1}){\phi_{i}}(z_{2}){{\phi_{i}}}(z_{3})\rangle=\frac{\mathcal{G}^{TT\phi\phi}(\eta)}{(z_{01})^{4}(z_{23})^{2h_{i}}}~{},$
$\displaystyle\mathcal{G}^{TT\phi\phi}(\eta)=\left(\frac{c}{2}+\frac{h_{i}\eta^{2}(\eta(h_{i}\eta-2)+2)}{(\eta-1)^{2}}\right)~{}.$
(A.44)

The cross-ratio $\eta$ is

$\displaystyle\eta\equiv\frac{z_{01}z_{23}}{z_{02}z_{13}}~{}.$ (A.45)

We can recursively apply the algorithm used above to add more insertions of
$T$. For example:

$\displaystyle\langle
T(z_{0})T(z_{1})T(z_{2}){\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle=$
$\displaystyle\mathcal{L}_{-2}^{(z_{0})}\langle
T(z_{1})T(z_{2}){\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle+\frac{c/2}{(z_{01})^{4}}\langle
T(z_{2}){\mathcal{X}}^{(0,n)}(\\{z_{i}\\})\rangle+\frac{c/2}{(z_{02})^{4}}\langle
T(z_{1}){\mathcal{X}}^{(0,n)}(z_{i})\rangle~{}.$ (A.46)

As a particular case of eq. (A.3.1) , when ${\mathcal{X}}^{(0,n)}$ is the
identity operator and using

$\displaystyle\langle
T(z_{2})T(z_{3})\rangle=\frac{c/2}{(z_{23})^{4}}~{},\quad\langle
T(z_{1})T(z_{2})T(z_{3})\rangle=\frac{c}{(z_{12})^{2}(z_{23})^{2}(z_{31})^{2}}~{},$
(A.47)

we find the four-point function of $T(z)$

$\displaystyle\langle
T(z_{0})T(z_{1})T(z_{2})T(z_{3})\rangle=\frac{\mathcal{G}^{TTTT}(\eta)}{(z_{01})^{4}(z_{23})^{4}}~{},$
$\displaystyle\mathcal{G}^{TTTT}(\eta)$
$\displaystyle=\frac{2c~{}\eta^{2}((\eta-1)\eta+1)}{(\eta-1)^{2}}+\frac{c^{2}}{4}\left(1+\eta^{4}+\left(\frac{\eta}{\eta-1}\right)^{4}\right)~{}.$
(A.48)

The four-point correlation function of $T(z)$ is a physical correlator, and as
such it admits a decomposition into the $SL(2,\mathbb{R})$ conformal blocks of
eq. (A.6) (with positive coefficients in unitary theories). In the $s$-channel
we find

$\displaystyle\mathcal{G}^{TTTT}(\eta)=\frac{c^{2}}{4}G(0,0,0,\eta)+\sum_{n=2,4,\dots}C_{TT}{}^{n}C_{TTn}\,G(0,0,n,\eta)~{},$
(A.49)

with squared OPE coefficients

$\displaystyle
C_{TT}{}^{n}C_{TTn}=\frac{\sqrt{\pi}c(c(n-3)(n-2)(n-1)\Gamma(n+3)+288((n-1)n-1)\Gamma(n))}{9\times
2^{2n+1}\Gamma\left(n-\frac{1}{2}\right)}~{}.$ (A.50)

In particular, the squared OPE coefficients with $T$ and $T^{2}$ in the
$T\times T$ OPE read

$\displaystyle C_{TT}{}^{T}C_{TTT}=2c~{},\quad
C_{TT}{}^{T^{2}}C_{TTT^{2}}=\frac{c}{10}(22+5c)~{}.$ (A.51)

Note that eq. (A.40) implies that $C_{TT}{}^{T^{2}}=1$, and so from the
rightmost equation above we get

$\displaystyle\langle T^{2}(0)T^{2}(\infty)\rangle\equiv
C_{T^{2}}=C_{TTT^{2}}=\frac{c}{10}(22+5c)~{}.$ (A.52)

####  A.3.2 Correlation functions of the stress tensor

Let us now return to the BCFT setup. We recall that $n$-point functions of the
stress tensor $T(z)$ on the upper half-plane are the same as in flat space.
The reason for this is holomorphy, which states that

$\displaystyle\bar{\partial}_{i}\langle T(z_{1})T(z_{2})\dots
T(z_{n})\rangle_{{\mathbb{H}^{+}}}=0~{}.$ (A.53)

and so the correlation functions are invariant under a translation with
respect to the transverse coordinate orthogonal to the boundary. Therefore,
they must take the same form as in the full complex plane [40, 41, 42]. For
two- and three-point correlation functions of $T$, the bulk self-OPE of $T$
together with $SL(2,\mathbb{R})$ symmetry dictate that

$\displaystyle\langle
T(z_{1}){T}(z_{2})\rangle_{{\mathbb{H}^{+}}}=\frac{c/2}{(z_{12})^{4}}~{},\quad\langle
T(z_{1}){T}(z_{2})T(z_{3})\rangle_{{\mathbb{H}^{+}}}=\frac{c}{(z_{12})^{2}(z_{23})^{2}(z_{31})^{2}}~{}.$
(A.54)

Analogous results hold for higher-point correlation functions of $T$ or of
$\bar{T}$. Note that eq. (A.16) implies that correlation functions of
$\bar{T}$ are those of $T$ on the upper half-plane, analytically continued to
the lower half-plane, so for example

$\displaystyle\langle T(z)\bar{T}(\bar{w})\rangle_{{\mathbb{H}^{+}}}=\langle
T(z){T}(w^{*})\rangle_{{\mathbb{H}^{+}}}~{}.$ (A.55)

####  A.3.3 Correlation functions of the displacement operator

The $n$-point functions of the displacement operator are directly obtained
from eq. (A.54), by simply restricting all $T$-insertions to the real line. We
choose an operator ordering along the line so that $x_{i}>x_{i+1}$, and so we
find

$\displaystyle\langle\text{D}(x_{1})\text{D}(x_{2})\rangle_{{\mathbb{H}^{+}}}=\frac{\hat{C}_{\text{D}}}{x_{12}^{4}}~{},$
$\displaystyle\quad\langle\text{D}(x_{1})\text{D}(x_{2})\text{D}(x_{3})\rangle_{{\mathbb{H}^{+}}}=\frac{\hat{C}_{\text{D}\text{D}\text{D}}}{(x_{12})^{2}(x_{23})^{2}(x_{31})^{2}}~{},$
$\displaystyle\hat{C}_{\text{D}}=c/2~{},\quad\hat{C}_{\text{D}\text{D}\text{D}}=c~{}.$
(A.56)

Similarly, the four-point correlation function can be obtained from eq.
(A.3.1), e.g.

$\displaystyle\langle\text{D}(x_{1})\text{D}(x_{2})\text{D}(x_{3})\text{D}(x_{4})\rangle_{{\mathbb{H}^{+}}}=\frac{\mathcal{G}^{\text{D}\text{D}\text{D}\text{D}}(\eta)}{(x_{12})^{4}(x_{34})^{4}}~{},\quad
x_{i}>x_{i+1},$
$\displaystyle\mathcal{G}^{\text{D}\text{D}\text{D}\text{D}}(\eta)=\frac{2c~{}\eta^{2}((\eta-1)\eta+1)}{(\eta-1)^{2}}+\frac{c^{2}}{4}\left(1+\eta^{4}+\left(\frac{\eta}{\eta-1}\right)^{4}\right)~{}.$
(A.57)

The cross-ratio $\eta$, which can be found in eq. (A.4), is the restriction of
the familiar holomorphic cross-ratio in eq. (A.45) to the real line. We can
decompose this correlator into the $s$-channel $SL(2,\mathbb{R})$ blocks
defined in eq. (A.3.3). The resulting squared OPE coefficients equal the ones
of eq. (A.49), so in particular from eq. (A.51) we find

$\displaystyle\hat{C}_{\text{D}\text{D}}{}^{\text{D}}\hat{C}_{\text{D}\text{D}\text{D}}=2c~{},\quad\hat{C}_{\text{D}\text{D}}{}^{\text{D}^{2}}\hat{C}_{\text{D}\text{D}\text{D}^{2}}=\frac{c}{10}(22+5c)~{}.$
(A.58)

In the equation above we identified the global primary operator $\text{D}^{2}$
as the boundary limit of the operator $T^{2}$ in the bulk, which is defined in
eq. (A.40). From eq. (A.52) we find

$\displaystyle\langle\text{D}^{2}(0)\text{D}^{2}(\infty)\rangle_{{\mathbb{H}^{+}}}\equiv\hat{C}_{\text{D}^{2}}=\frac{c}{10}(22+5c)~{},$
(A.59)

and, combining with eq. (A.58)

$\displaystyle\hat{C}_{\text{D}\text{D}}{}^{\text{D}^{2}}=1~{},\quad\hat{C}_{\text{D}\text{D}\text{D}^{2}}=\hat{C}_{\text{D}^{2}}.$
(A.60)

####  A.3.4 Correlation functions of the displacement with boundary operators

Let us now consider a conformal boundary condition that allows for a boundary
Virasoro primary $\psi$ with two-point correlation function

$\displaystyle\langle\psi(x_{1})\psi(x_{2})\rangle_{{\mathbb{H}^{+}}}=\frac{1}{x_{12}^{2\hat{\Delta}}}~{},\quad
x_{1}>x_{2}~{}.$ (A.61)

As we discussed in the previous subsection, as far as Ward identities are
concerned, boundary Virasoro primaries behave as holomorphic Virasoro
primaries. Correlation functions between D and $\psi$ can be obtained from
correlation functions on the upper half-plane between $T$ and a holomorphic
Virasoro primary $\phi$ with weight $h=\hat{\Delta}$. From eq. (A.43), upon
using the bulk-boundary OPE of $T$ in eq. (A.17) and choosing the operator
ordering $x_{i}>x_{i+1}$ on the line we immediately get

$\displaystyle\langle\text{D}(x_{1})\psi(x_{2})\psi(x_{3})\rangle_{{\mathbb{H}^{+}}}$
$\displaystyle=\frac{\hat{C}_{\psi\psi\text{D}}}{(x_{12})^{2}(x_{23})^{2\hat{\Delta}-2}(x_{31})^{2}}~{},\quad\hat{C}_{\psi\psi\text{D}}=\hat{C}_{\text{D}\psi\psi}=\hat{\Delta}~{}.$
(A.62)

This result, when combined with eq. (A.3.3), tells us that the displacement