###### Abstract

We study the two-point function of local operators in the critical $O(N)$
model in the presence of a magnetic field localized on a line. We use a
recently developed conformal dispersion relation to compute the correlator at
first order in the $\epsilon$-expansion and we extract the full set of defect
and bulk CFT data using the Lorentzian inversion formulae. The only input for
the computation of the connected correlator is its discontinuity at first
order in perturbation theory, which is determined by the anomalous dimension
of a single bulk operator. We discuss possible low-spin ambiguities and
perform several diagrammatic checks of our results.

Analytic bootstrap for the localized magnetic field

Lorenzo Bianchia,b<EMAIL_ADDRESS>Davide Bonomic
<EMAIL_ADDRESS>Elia de Sabbataa,b<EMAIL_ADDRESS>

a Dipartimento di Fisica, Università di Torino and INFN - Sezione di Torino
Via P. Giuria 1, 10125 Torino, Italy

b I.N.F.N. - sezione di Torino,
Via P. Giuria 1, I-10125 Torino, Italy

c Department of Mathematics, City, University of London,
Northampton Square, EC1V 0HB London, United Kingdom

###### Contents

  1. 1 Introduction and discussion
  2. 2 The O(N) critical model with a localized magnetic field
    1. 2.1 The observable: bulk two-point function
    2. 2.2 Block expansions
  3. 3 Lorentzian inversion formulae and dispersion relation
  4. 4 Tree level
  5. 5 One loop
    1. 5.1 The full result
    2. 5.2 Defect channel data
    3. 5.3 Bulk channel data
  6. A Kinematics and conformal blocks
  7. B Explicit computation of the one-loop correlator
    1. B.1 Generating function of the bulk inversion
  8. C Feynman diagrams
    1. C.1 Evaluation of the integral for $z=\bar{z}$
    2. C.2 Evaluation of the integral for $z\bar{z}=1$

##  1 Introduction and discussion

The development of conformal bootstrap methods, both numerical and analytical,
had an extraordinarily strong impact on the study of statistical models. One
of the most exciting progress has certainly been the numerical estimation of
the critical exponents of several statistical systems with unprecedented
precision [1, 2, 3, 4, 5]. In parallel, the discovery of powerful analytical
techniques, such as the Lorentzian inversion formula [6, 7] and the conformal
dispersion relation [8], allowed us to make progress in situations where the
conformal field theory contains a small parameter (not necessarily the
coupling). With these methods one can obtain results order by order in the
expansion parameter only using symmetries and internal consistency.
Specifically, for the case of critical models in $(4-\epsilon)$ dimensions,
analytic bootstrap techniques can be successfully used to reproduce and extend
the results in $\epsilon$-expansion [9, 10, 11, 12, 13].

While a large wealth of results is now available for correlators of local
operators, the study of extended excitations, or defects, with bootstrap
techniques has just started to reveal some of its interesting features [14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
53, 54, 55]. In particular, compared to the homogeneous case, where the
crossing equation relates different OPE channels of local operators, the
crossing equation for a bulk two-point function in the presence of a defect
allows for a cross-talk between two very different OPE expansions [56]. In the
bulk channel the two operators are expanded in the bulk OPE, i.e. as infinite
sum over local bulk operators, while in the defect channel both operators are
expanded in terms of an infinite tower of defect operators. This important
property has led to the formulation of two different Lorentzian inversion
formulae: one that reconstructs the defect spectrum through a discontinuity
that is controlled by the bulk OPE [57] and one that does the opposite [58].
Starting from these inversion formulae, whose purpose is to extract the CFT
data starting from the (double-)discontinuity of the correlator, one can
derive conformal dispersion relations reconstructing the full correlator from
its (double-)discontinuity [59, 60].

In this paper we apply these techniques to study the two-point correlator of
local operators in the $O(N)$ critical model with a localized magnetic field.
The $O(N)$ critical model is a well-known conformal field theory which can be
realized by deforming the four-dimensional free theory of $N$ scalar fields by
a quartic $O(N)$ invariant interaction tuning the coupling to the Wilson-
Fisher fixed point in $(4-\epsilon)$ dimension. Famous applications, i.e.
critical models in the same universality class, are the Ising model for $N=1$,
the XY model and the Helium superfluid transition for $N=2$ and the isotropic
magnets for $N=3$.

A natural extended excitation in this theory is obtained by switching on a
magnetic field along a line, breaking the $O(N)$ symmetry down to $O(N-1)$
along the defect. This line defect can be nicely realized on the lattice and
it has been studied with Monte Carlo simulations in [61, 62]. Also
experimental applications are conceivable, either in quantum simulators [63]
or in a mixture of two liquids with a colloidal impurity [64, 65, 62]. It is
therefore important to produce predictions for the defect CFT data of this
critical system. Field-theoretical studies for some observables either at
large $N$ or in $\epsilon$-expansion are available in [66, 67, 68]. In
particular, the recent study of [68], besides providing new results for some
observables in the $\epsilon$-expansion, provided a well-defined strategy to
study the large $N$ regime finding consistent results in the two expansions.
Even more recently [69] analyzed this defect for $N=3$ using numerical
bootstrap techniques for the four-point function of defect operators.

Here we initiate the analysis of bulk two-point functions, which give access
to an infinite set of defect CFT data. The techniques we use seem to rely very
little on the specific defect under analysis. Therefore, it is very likely
that similar methods could be applied to other line defects in the $O(N)$
critical theory, such as the magnetic impurities considered in [70, 71, 72,
73, 74, 75, 76, 77] or the twist defects [78, 79]. Very recently, [80]
introduced a class of conformal line defects in fermionic Gross-Neveu-Yukawa
models and it would be very interesting to extend our analysis to that case as
well.

### Summary of the results

For the $O(N)$ Wilson Fisher theory in $3<d<4$ the fundamental excitation is
the vector of scalar fields $\phi_{i}$ with $i=1,\dots,N$. The main observable
in this work is the two-point function

$\langle\phi_{i}(x)\phi_{j}(y)\rangle_{D},$ (1.1)

in the presence of the line defect $D$ obtained by coupling the field
$\phi_{1}$ to a magnetic field localized on a line. At the critical point the
theory is described by a defect CFT and the bulk correlator (1.1) is fixed up
to two functions of the conformal cross ratios $z$ and $\bar{z}$

$\langle\phi_{i}(x)\phi_{j}(y)\rangle_{D}=\frac{\hat{F}_{S}(z,\bar{z})\delta_{i1}\delta_{j1}+\hat{F}_{V}(z,\bar{z})\left(\delta_{ij}-\delta_{i1}\delta_{j1}\right)}{|x_{\perp}|^{\Delta_{\phi}}|y_{\perp}|^{\Delta_{\phi}}}.$
(1.2)

In $\epsilon$-expansion, the functions $\hat{F}_{S}$ and $\hat{F}_{V}$ can be
expanded perturbatively and one of the goals of this work is to determine them
at first order in $\epsilon$.

Diagrammatically, one can easily see that there is only one connected Feynman
diagram contributing to these functions at that order. Therefore, it does not
seem to be necessary to appeal to the bootstrap machinery for this
computation. Nevertheless, a more careful analysis shows that the diagram is
surprisingly hard to compute with ordinary techniques and it does not take
values in the space of generalized polylogarithmic functions. Indeed, even its
discontinuity is expressed in terms of elliptic functions. From a bootstrap
standpoint, this complicated structure has a very clear origin. The non-
trivial part of the discontinuity of the functions $\hat{F}_{S}$ and
$\hat{F}_{V}$ comes from a single bulk conformal block, associated to the
lightest operator exchanged in the bulk channel. The bulk conformal blocks for
a line defect in four dimensions are not known in a closed form, but for the
specific values $\Delta=2$ and $\ell=0$ that are relevant in this case we can
write down the block (and consequently the discontinuity) in terms of
incomplete elliptic integrals of the first kind.

Despite the complicated functional form, the fact that a single bulk operator
completely determines the discontinuity is an important conceptual step.
Indeed, the dispersion relation developed in [59, 60] allows to reconstruct
the full correlator from its discontinuity. Since the anomalous dimension of
bulk operators are known from the analysis of the homogeneous bulk theory, we
can reconstruct the non-trivial part of the one-loop correlator using as our
only input a single piece of bulk CFT data. This is a universal feature of
defects in the $O(N)$ critical model since it does not require any knowledge
of the specifics of the defect. However, the differences among defects may
arise in the contributions at low transverse spin. Indeed, the Lorentzian
inversion formula may fail to reproduce low spin data if the correlator is not
sufficiently well-behaved in a particular limit. Correspondingly, the
dispersion relation may lead to a result which misses (possibly infinitely
many) low spin contributions. For the case of the localized magnetic field we
show by comparison to Feynman diagrams that this low spin problem affects the
result very mildly and the only additional piece of CFT data that we need is
the one-loop one-point function of the fundamental field $\phi$ which provides
a disconnected contribution and was computed in [67, 68]. It would be
interesting to explore how this low-spin ambiguity is important for other line
defects in the $O(N)$ critical model.

The full result for the correlator (1.1) takes a complicated form, which we
spell out in (5.15). Here it is more interesting to show the results for the
CFT data of the exchanged operators. In the defect channel, the exchanged
operators transform in irreducible representations of the preserved $O(N-1)$
symmetry and in the correlator (1.1) singlet (S) and vector (V) operators are
exchanged. In particular, only the family of operators that already appeared
at tree level contribute at one loop because the bulk-to-defect couplings of
other operators are suppressed by a power $\epsilon^{2}$. Therefore, we
extracted the one-loop anomalous dimensions $\gamma^{(1)}$ and squared bulk-
to-defect couplings $\hat{b}^{2(1)}$ as a function of the transverse spin $s$.

$\displaystyle\hat{\gamma}_{S,0,s}^{(1)}=\frac{1-s}{(2s+1)},$
$\displaystyle\hat{b}_{S,0,s}^{2(1)}=\frac{-2(s-1)H_{s}-3H_{s+\frac{1}{2}}}{2(2s+1)},$
$\displaystyle\hat{\gamma}_{V,0,s}^{(1)}=-\frac{s}{(2s+1)},$
$\displaystyle\hat{b}_{V,0,s}^{2(1)}=-\frac{(2s+1)\left(2sH_{s}+H_{s-\frac{1}{2}}\right)+2}{2(2s+1)^{2}},$
(1.3)

where $H_{z}$ is the harmonic number. Notice the presence of zeroes in the
anomalous dimensions for $s=1$ in the singlet channel and for $s=0$ in the
vector channel. These values correspond to the displacement and the tilt
operators respectively. The former is a protected defect operator associated
to the explicit breaking of translation invariance, while the latter is a
$O(N-1)$ vector associated to the $N-1$ broken generators of the internal
symmetry. Therefore, this is an important consistency check of our result.

Knowing the full result one can also extract the defect CFT data appearing in
the bulk channel. In that case, the exchanged operators transform in the
singlet (S) or in the symmetric traceless (T) representation of $O(N)$. Of
course, the anomalous dimensions are already known since they are not affected
by the defect. The new data are the one-loop values for the product of the
one-point function of the exchanged operator $a_{\mathcal{O}}$ and the bulk
OPE coefficients $\lambda_{\phi\phi\mathcal{O}}$. For the twist-two operators
that already appeared at tree level we find the correction

$\displaystyle
a\lambda^{(1)}_{T,0,\ell}=Na\lambda^{(1)}_{S,0,\ell}=-\frac{2^{-\ell-7}\Gamma\left(\frac{\ell}{2}+\frac{1}{2}\right)^{3}(N+8)}{\pi\Gamma\left(\frac{\ell}{2}+1\right)\Gamma\left(\ell+\frac{1}{2}\right)}\times$
$\displaystyle\times\Big{(}-32H_{\frac{\ell}{2}-\frac{1}{2}}+35H_{\ell-\frac{1}{2}}+19\psi^{(0)}(\ell)-38\psi^{(0)}(2\ell)-19\gamma+38\log(2)+16\frac{N^{2}-3N-22}{(N+8)^{2}}\Big{)},$
(1.4)

where $\psi^{(0)}(z)$ is the digamma function. In principle, also twist-four
operators can be exchanged. Nevertheless, while at twist two there is only a
single family of operators in the OPE of two fundamental fields $\phi_{i}$,
starting from twist four one can create more than one primary for a given
number of fields and derivatives. These operators would enter the OPE with
their classical dimensions and therefore they are degenerate at this order in
perturbation theory. To lift the degeneracy one would need to compute
additional correlators. Still, one can extract the quantities
$a\lambda^{(1)}_{S,1,\ell}$ and $a\lambda^{(1)}_{T,1,\ell}$ (our results are
given in (5.34)), which correspond, for the degenerate cases, to linear
combinations of the defect CFT data associated to the single degenerate
operators. Furthermore, since the bulk OPE coefficients are already at order
$\epsilon$, the only new information is about the tree-level one-point
functions of twist-four operators, which can be easily obtained also by other
means.

The plan of the paper is as follows: in Section 2 we review the $O(N)$
critical model in the presence of a localized magnetic field, in Section 3 we
briefly present the dispersion relation and Lorentzian inversion formulae,
which we use in Section 4 and 5 to compute the correlator and extract the
defect CFT data at tree level and at one loop, respectively. Some technical
details and diagrammatic checks are contained in the Appendix.

Note added: While this paper was in preparation, we became aware of [81],
whose content partially overlaps with the present work. We coordinated with
the author for a simultaneous submission.

##  2 The O(N) critical model with a localized magnetic field

The starting point for the construction of the $O(N)$ critical model is the
Euclidean action of $N$ free massless scalar fields in the fundamental
representation of $O(N)$ perturbed by a $O(N)$-invariant quartic interaction

$S=\int
d^{d}x\left[\frac{1}{2}\left(\partial_{\mu}\phi_{i}\right)^{2}+\frac{\lambda_{0}}{4!}\left(\phi_{i}\phi_{i}\right)^{2}\right],$
(2.1)

where $d=4-\epsilon$. Since $d<4$ the perturbation is relevant and it triggers
a renormalization group flow. This flow can be studied by working
pertirbatibvely in $\lambda_{0}$; in particular we adopt the minimal
subtraction (MS) scheme. The beta function of the renormalized coupling
constant $\lambda$ at one loop reads

$\beta(\lambda)=\frac{\partial\lambda}{\partial\log\mu}=-\epsilon\lambda+\frac{N+8}{48\pi^{2}}\lambda^{2}+O(\lambda^{3}),$
(2.2)

where $\mu$ is the mass scale introduced in the renormalization process. If
the mass term is fine-tuned to zero, this flow admits the well known infrared
Wilson-Fisher fixed point [82], which describes a conformal field theory. This
fixed point corresponds to the non-trivial zero of the beta function

$\beta(\lambda_{*})=0\quad\Rightarrow\quad\lambda_{*}=\frac{48\pi^{2}}{N+8}\epsilon+\mathcal{O}(\epsilon^{2}).$
(2.3)

In the perturbative setup, the operators of this conformal field theory are
just the renormalized version of the local operators constructed using the
bare fields. For example, the renormalized field $[\phi_{i}](x)$ is given by

$[\phi_{i}](x)=Z_{\phi}\,\phi_{i}(x),$ (2.4)

where in this case the wavefunction renormalization $Z_{\phi}$ is just a
factor since there is no operator mixing. The wavefunction renormalization can
be computed by imposing the finiteness of the correlators. In particular, at
one loop it is readily found that

$Z_{\phi}(\lambda)=1+O(\lambda^{2}).$ (2.5)

From the wavefunction renormalization it is possible to extract the anomalous
dimension of the field $\phi_{i}$ in the following way

$\gamma_{\phi}(\lambda)=-\frac{\partial\log
Z_{\phi}}{\partial\log\mu}=-\beta(\lambda)\frac{\partial\log
Z_{\phi}}{\partial\lambda}=0+O(\lambda^{2}).$ (2.6)

The conformal dimension $\Delta_{\phi}$ of the operator $[\phi_{i}]$ is [83]

$\Delta_{\phi}=\frac{d-2}{2}+\gamma_{\phi}(\lambda_{*})=1-\frac{\epsilon}{2}+\mathcal{O}(\epsilon^{2}).$
(2.7)

Note that at this order, the conformal dimension coincides exactly with the
engineering dimension of the bare field, since the anomalous dimension
vanishes. We can now consider the case in which we deform the action also by
an additional perturbation localized on a line

$S_{D}=h_{0}\int d\tau\,|\dot{x}(\tau)|\phi_{1}(x(\tau)),$ (2.8)

where $x(\tau)$ describes a line as the real parameter $\tau$ varies, which we
denote by $D$ (called defect), and $h_{0}$ is a new coupling constant. In
particular, we will consider the case in which $D$ is a straight line (or
equivalently, a circle). This perturbation explicitly breaks the $O(N)$ global
symmetry of the model down to $O(N-1)$. For the free theory in four
dimensions, this perturbation produces a simple example of a conformal defect
as the operator $\phi$ has dimension one and $h_{0}$ is a defect marginal
parameter [84]. As we move away from four dimension, the bulk theory flows to
the Wilson-Fisher fixed point and the operator $\phi$ is a weakly relevant
defect deformation (the bulk dimension is $4-\epsilon$, but the defect
dimension is fixed to one). Therefore, this perturbation together with the
quartic interaction triggers a renormalization group flow in the two coupling
constants. One can study this joint flow using standard diagrammatic
techniques without the need to work perturbatively in $h_{0}$, since any
diagram contributing to any correlator at some fixed order in $\lambda$ will
contain insertion of $h_{0}$ only up to a finite power. That is to say,
$h_{0}$ is not considered to be a small coupling constant. This flow admits an
infrared fixed point for the following value444 As one can find in [68], to
calculate the value of $h_{*}$ to the first order in $\epsilon$ one need to
consider perturbations up to two loops. However, for the purpose of this work
the small parameter entering in the analytic bootstrap is $\epsilon$, hence we
do not truncate results at one or two loops but rather at order one in
$\epsilon$.  of the renormalized coupling constants [67, 68]

$\lambda_{*}=\frac{48\pi^{2}}{N+8}\epsilon+\mathcal{O}(\epsilon^{2}),\quad
h_{*}=\sqrt{N+8}+\frac{4N^{2}+45N+170}{4(N+8)^{\frac{3}{2}}}\epsilon+\mathcal{O}(\epsilon^{2}).$
(2.9)

The defect perturbation breaks explicitly the conformal symmetry group
$SO(d+1,1)$ of the system at the fixed point. Since we have chosen the defect
$D$ to be a straight line, the system will still be invariant under a residual
symmetry group $SO(2,1)\times SO(d-1)$, generated by conformal transformation
on the line and rotation around the defect $D$. This is equivalent to saying
that the fixed point under consideration can be described by a (line) defect
conformal field theory.

###  2.1 The observable: bulk two-point function

The main observables of this defect conformal field theory (dCFT) are
correlators of local operators in the presence of the defect. From this
section onwards every local operator will be assumed to be already
renormalized. There are two kind of local operators: bulk operators and defect
operators. Bulk operators are well-defined in those points of the space which
do not lie in the defect $D$, whereas defect operators have $D$ as their
support. In the $\epsilon$-expansion, both kinds of operators arise as the
renormalized version of composite operators evaluated at the fixed point. The
residual conformal symmetry group $SO(2,1)\times SO(d-1)$ imposes severe
constraints on the correlators through Ward identities analogous to the
homogeneous case. The constraints are of course weaker due to the presence of
the defect $D$. For instance, one-point functions of generic bulk operators
$O_{\Delta,\mu_{1}\dots\mu_{\ell}}(x)$ do not vanish, but instead they take
the form [56]

$\langle\mathcal{O}_{\Delta,\mu_{1}\dots\mu_{\ell}}(x)\rangle_{D}=\frac{a_{\mathcal{O}}}{|x_{\perp}|^{\Delta}}I_{\mu_{1}\dots\mu_{\ell}}(x_{\perp}),$
(2.10)

where $x_{\perp}$ is the projection of $x$ in the subspace orthogonal to the
defect $D$, $\Delta$ is the conformal dimension of the operator and
$I_{\mu_{1}\dots\mu_{\ell}}(x_{\perp})$ is a completely determined tensor555
The specific form of this tensor $I_{\mu_{1}\dots\mu_{l}}(x_{\perp})$ can be
easily determined using the embedding formalism, see e.g. [56]. . The
constants $a_{\mathcal{O}}$ are real and they are part of the data of the
dCFT. In our model, there is an additional global $O(N-1)$ symmetry which
imposes further constraints on the correlators. For example, the one-point
function of $\phi_{i}(x)$ must be proportional to the only $O(N-1)$ invariant
tensor with one free index, namely $\delta_{i1}$,

$\langle\phi_{i}(x)\rangle_{D}=\delta_{i1}\frac{a_{\phi}}{|x_{\perp}|^{\Delta_{\phi}}}.$
(2.11)

A much more interesting correlator, on which this work will focus, is the bulk
two-point function of two fundamental fields

$\langle\phi_{i}(x)\phi_{j}(y)\rangle_{D},\quad\quad x,y\notin D.$ (2.12)

Conformal and global symmetries constrain this correlator to take the
following form

$\langle\phi_{i}(x)\phi_{j}(y)\rangle_{D}=\frac{F_{1}(z,\bar{z})\delta_{ij}+F_{2}(z,\bar{z})\delta_{i1}\delta_{j1}}{|x_{\perp}|^{\Delta_{\phi}}|y_{\perp}|^{\Delta_{\phi}}},$
(2.13)

where $F_{1}(z,\bar{z})$ and $F_{2}(z,\bar{z})$ are arbitrary functions of the
two conformally invariant cross-ratios $z$ and $\bar{z}$ defined by

$\frac{(x-y)^{2}}{|x_{\perp}||y_{\perp}|}=\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}},\quad\frac{x\cdot
y}{|x_{\perp}||y_{\perp}|}=\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}.$ (2.14)

The geometrical interpretation of $z$ and $\bar{z}$ becomes transparent if we
use conformal transformations to put the two bulk points on a plane orthogonal
to the defect and then to move one of the two to $(1,0)$ on this plane. Then,
in Euclidean signature, $z$ and $\bar{z}$ are complex coordinates for the
second point on this plane. It is also useful to introduce radial coordinates
$r$ and $w$ on the orthogonal plane

$\quad z=rw,\quad\bar{z}=\frac{r}{w}.$ (2.15)

Rotating to Lorentzian signature with time orthogonal to the defect, $z$ and
$\bar{z}$ become real and independent lightcone coordinates.

It is well known that in a CFT there exists an operator product expansion
(OPE) for any couple of primary operators. This OPE is convergent inside
correlation functions provided that there are no other operator insertions
closer to the point of expansion than the two primaries. In a dCFT, the same
OPE of the bulk theory holds with the additional condition that the defect
also has to be sufficiently distant from the point of expansion. In addition
to the bulk OPE however, there is also a new channel, called defect channel
[56]. It consists of an expansion of a bulk operator as an infinite sum of
defect operators, and it is convergent inside correlation functions whenever
the operator is close enough to the defect. In general, these two expansions
can be further refined to manifestly account for the structure of the internal
symmetry of the theory.

For the bulk channel, the OPE of two operators transforming under the
representations $R_{1}$ and $R_{2}$ of the internal symmetry group must
contain exchanged operators in an irreducible representation contained in the
Clebsch-Gordan decomposition of $R_{1}\otimes R_{2}$. In the case of our
model, for two fundamental scalar fields it is possible to write symbolically

$\phi_{i}(x)\phi_{j}(y)=\sum_{\Delta,\ell,R}\,\lambda_{\phi\phi\mathcal{O}}^{ij\,a}\,|x-y|^{\Delta-2\Delta_{\phi}}\left(\mathcal{O}_{\Delta,\ell}^{\,a}(y)+\text{descendants}\right),$
(2.16)

where $\Delta_{\phi}$ is the conformal dimension of the field $\phi$. The
conformal dimension and the spin of the exchanged primary operator
$\mathcal{O}_{\Delta,\ell}^{\,a}$ are labelled by $\Delta$ and $\ell$
respectively, and $R$ denotes the $O(N)$ irreducible representations contained
in the tensor product $V\otimes V$ of two vector representations V, i.e. the
singlet $S$, the symmetric traceless $T$ and the antisymmetric $A$. The index
$a$ runs over $a=1,\dots,\text{dim}R$. In (2.16) we have suppressed spacetime
indices. Since this is a OPE between scalar operators, only operators in even
spin-$\ell$ traceless symmetric representations of $SO(d)$ can appear in the
decomposition. The tensor structure of the three-point function coefficients
$\lambda_{\phi\phi\mathcal{O}_{S}}^{ij}$ can be easily written down for
$R=S,T,A$ as

$\begin{split}\lambda_{\phi\phi\mathcal{O}_{S}}^{ij}=\lambda_{\phi\phi\mathcal{O}_{S}}\delta_{ij},\quad\lambda_{\phi\phi\mathcal{O}_{T}}^{ij\,(kl)}=\lambda_{\phi\phi\mathcal{O}_{T}}\left(\delta_{i(k}\delta_{jl)}-\frac{1}{N}\delta_{ij}\delta_{kl}\right),\quad\lambda_{\phi\phi\mathcal{O}_{A}}^{ij\,[kl]}=\lambda_{\phi\phi\mathcal{O}_{A}}\delta_{i[k}\delta_{jl]},\end{split}$
(2.17)

where the (anti-)symmetrization has been taken with weight $1/2$.

On the other hand, the operators appearing in the defect OPE will be organized
into $O(N-1)$ representations since the $O(N)$ symmetry is explicitly broken
by the defect. In general, if the symmetry group $G$ is broken down to a
subgroup $H$ by the defect and if a bulk operator sits in the representation
$R$ of $G$, the operators exchanged in the defect OPE must transform in an
irreducible representations contained in the branching rule of the restricted
representation $R^{(G)}\rightarrow R^{(H)}$. Hence we can write

$\phi_{i}(x)=\sum_{\hat{\Delta},s,R}b^{\,i\,a}_{\phi\hat{\mathcal{O}}}|x_{\perp}|^{\hat{\Delta}-\Delta_{\phi}}\left(\hat{\mathcal{O}}^{\,a}_{\hat{\Delta},s}(x)+\text{descendants}\right),$
(2.18)

where $\hat{\Delta}$ and $s$ are the conformal dimension and the transverse
spin of the defect operator $\hat{\mathcal{O}}_{\hat{\Delta},s}^{\,a}$
respectively. Since $\phi_{i}(x)$ is a scalar, the operators exchanged in this
OPE do not carry longitudinal spin. From the branching rule
$V^{O(N)}\rightarrow S^{O(N-1)}\oplus V^{O(N-1)}$ one can immediately see that
the only allowed representations $R$ are the singlet $S$ and the vector $V$ of
$O(N-1)$. For these two representations, the tensor structures of the bulk-
defect two-point function coefficients $b^{\,i\,a}_{\phi\hat{\mathcal{O}}}$
are

$b^{\,i}_{\phi\hat{\mathcal{O}}_{S}}=b_{S,\hat{\Delta},s}\delta_{i1},\quad\quad
b^{\,i\,\hat{j}}_{\phi\hat{\mathcal{O}}_{V}}=b_{V,\hat{\Delta},s}\delta_{i\hat{j}},$
(2.19)

where $\hat{j}=2,\dots,N$, and $\delta_{i1}$ and $\delta_{i\hat{j}}$ are
projectors from the representation space of $V^{O(N)}$ to those of
$S^{O(N-1)}$ and $V^{O(N-1)}$ respectively.

These two different OPE decompositions lead to two conformal block expansions,
which we analyze in the following section.

###  2.2 Block expansions

To analyze the bulk block expansion, it is convenient to rewrite the bulk two-
point function as

$\langle\phi_{i}(x)\phi_{j}(y)\rangle_{D}=\frac{F_{S}(z,\bar{z})\delta_{ij}+F_{T}(z,\bar{z})\left(\delta_{i1}\delta_{j1}-\frac{1}{N}\delta_{ij}\right)}{|x_{\perp}|^{\Delta_{\phi}}|y_{\perp}|^{\Delta_{\phi}}},$
(2.20)

where $F_{S}(z,\bar{z})$ and $F_{T}(z,\bar{z})$ are linear combinations of the
functions $F_{1}(z,\bar{z})$ and $F_{2}(z,\bar{z})$ introduced in (2.13) (see
(A.4)). In terms of lightcone coordinates, this bulk channel decomposition
reads

$F_{S}(z,\bar{z})\delta_{ij}+F_{T}(z,\bar{z})\left(\delta_{i1}\delta_{j1}-\frac{1}{N}\delta_{ij}\right)=\left(\frac{\sqrt{z\bar{z}}}{(1-z)(1-\bar{z})}\right)^{\Delta_{\phi}}\\!\\!\\!\sum\limits_{\begin{subarray}{c}\Delta,\ell\\\
R=S,T\end{subarray}}\lambda_{\phi\phi\mathcal{O}}^{ij\,a}\,a_{\mathcal{O}}^{\,a}\,f_{\Delta,\ell}(z,\bar{z}),$
(2.21)

where the explicit form of the bulk conformal blocks
$f_{\Delta,\ell}(z,\bar{z})$ was found in [85] and it is given in Appendix A.
Note that the only bulk operators which can have a non vanishing one-point
function are those for which the identity appears as an exchanged operator in
their defect OPE. Therefore, the allowed tensor structures for the
coefficients $a_{\mathcal{O}}^{\,a}$ are obtained by projecting $O(N)$
representations into the singlet $S^{O(N-1)}$. It immediately follows that
operators in antisymmetric representations have zero one-point functions. For
this reason, the sum in (2.21) is taken only over the singlet $S$ and the
symmetric traceless $T$ representations of $O(N)$. The tensor structures of
the one-point functions for $R=T,S$ are

$\begin{split}a_{\mathcal{O}_{S}},\quad
a_{\mathcal{O}_{T}}^{(ij)}=a_{\mathcal{O}_{T}}\left(\delta_{i1}\delta_{j1}-\frac{1}{N}\delta_{ij}\right).\end{split}$
(2.22)

Inserting (2.17) and (2.22) into (2.21) one gets the following block
decompositions

$\begin{split}F_{S}(z,\bar{z})=\left(\frac{\sqrt{z\bar{z}}}{(1-z)(1-\bar{z})}\right)^{\Delta_{\phi}}\sum_{\Delta,\ell}\lambda_{\phi\phi\mathcal{O}_{S}}a_{\mathcal{O}_{S}}f_{\Delta,\ell}(z,\bar{z}),\\\
F_{T}(z,\bar{z})=\left(\frac{\sqrt{z\bar{z}}}{(1-z)(1-\bar{z})}\right)^{\Delta_{\phi}}\sum_{\Delta,\ell}\lambda_{\phi\phi\mathcal{O}_{T}}a_{\mathcal{O}_{T}}f_{\Delta,\ell}(z,\bar{z}).\end{split}$
(2.23)

In a similar fashion, for the defect channel it is helpful to rewrite the bulk
two-point function in the following way

$\langle\phi_{i}(x)\phi_{j}(y)\rangle_{D}=\frac{\hat{F}_{S}(z,\bar{z})\delta_{i1}\delta_{j1}+\hat{F}_{V}(z,\bar{z})\left(\delta_{ij}-\delta_{i1}\delta_{j1}\right)}{|x_{\perp}|^{\Delta_{\phi}}|y_{\perp}|^{\Delta_{\phi}}},$
(2.24)

where again $\hat{F}_{S}(z,\bar{z})$ and $\hat{F}_{V}(z,\bar{z})$ are linear
combinations of $F_{1}(z,\bar{z})$ and $F_{2}(z,\bar{z})$ given in (A.4). The
defect channel decomposition is

$\hat{F}_{S}(z,\bar{z})\delta_{i1}\delta_{j1}+\hat{F}_{V}(z,\bar{z})\left(\delta_{ij}-\delta_{i1}\delta_{j1}\right)=\sum\limits_{\begin{subarray}{c}\hat{\Delta},s\\\
R=S,V\end{subarray}}b^{\,i\,a}_{\phi\hat{\mathcal{O}}}b^{\,j\,a}_{\phi\hat{\mathcal{O}}}\hat{f}_{\hat{\Delta},s},$
(2.25)

where the explicit form of the defect conformal blocks
$\hat{f}_{\hat{\Delta},s}(z,\bar{z})$ is given in Appendix A. Using (2.19) one
gets

$\begin{split}\hat{F}_{S}(z,\bar{z})=\sum_{\hat{\Delta},s}b_{S,\hat{\Delta},s}^{2}\hat{f}_{\hat{\Delta},s},\\\
\hat{F}_{V}(z,\bar{z})=\sum_{\hat{\Delta},s}b_{V,\hat{\Delta},s}^{2}\hat{f}_{\hat{\Delta},s}.\end{split}$
(2.26)

Everything said so far holds at the non-perturbative level. However, more can
be said if one looks at the interplays between the block expansions (2.23) and
(2.26) with the perturbative series. For instance, consider the three-point
function coefficients $\lambda_{\phi\phi\mathcal{O}}$. If the operator
$\mathcal{O}$ is composed with an even666In the odd $2k+1$ case the
coefficient simply vanishes for the representation theory reasons stated in
the previous subsection. number $2(k+1)$ of fundamental fields, then a
straightforward diagrammatic argument immediately implies that this
coefficient is at least of order $k$ in $\epsilon$. In particular, to the
first order in $\epsilon$ only operators with $k=0$ or $k=1$ will enter in the
bulk block expansion of the functions $F_{S}(z,\bar{z})$ and
$F_{T}(z,\bar{z})$. Moreover, only the anomalous dimensions of operators with
$k=0$ are relevant at this order, since only the classical dimensions of
operators with $k=1$ will contribute to the expansions.

In the next sections it will be shown that from the analytic properties of the
correlator (2.13) it is possible to extract an infinite set of data about this
dCFT. In particular, we will be able to extract the product of one-point
function coefficients $a_{\mathcal{O}}$ and bulk three-point functions to the
first order in $\epsilon$ for an infinite family of operators, called twist-
two operators. The twist of a local operator
$O_{\Delta,\mu_{1}\dots\mu_{\ell}}(x)$ with dimension $\Delta$ and spin $\ell$
is defined as $\tau:=\Delta-\ell$. Twist-two operators (operators with
$\tau=2+O(\epsilon)$) are exactly the operators composed using two fundamental
fields that we mentioned above. When $\ell\geq 1$ they can be thought as the
renormalized version of the infinite weakly broken higher spin currents which
are conserved in the UV free theory. Their explicit expression has been found
in [86] and it is given by

$\begin{split}J^{a}_{\mu_{1}\dots\mu_{\ell}}(x)&=\mathcal{N}_{\ell}^{R}\,P^{\,a}_{ij}\,\sum_{n=0}^{\ell}c_{\ell\,n}\,\partial_{\\{\mu_{1}}\dots\partial_{\mu_{\ell-n}}\phi_{i}\,\partial_{\mu_{\ell-n+1}}\dots\partial_{\mu_{\ell}\\}}\phi_{j}(x),\\\
c_{\ell\,n}&=\frac{(-1)^{n}}{n!(\ell-n)!\Gamma(n+\frac{d}{2}-1)\Gamma(\ell-n+\frac{d}{2}-1)},\end{split}$
(2.27)

where brackets denote traceless symmetrization and $P^{\,a}_{ij}$ is a
projector into an irreducible representation $R$ of $O(N)$ labelled by the
index $a=1,\dots\dim R$; since it is made out of just two indices it can only
be $S$, $T$ or $A$. Moreover, from the form of the coefficients $c_{\ell\,n}$
one can easily check that for even (odd) $\ell$ the operator
$J^{a}_{\mu_{1}\dots\mu_{\ell}}(x)$ is (anti-)symmetric in the indices $i$ and
$j$, hence it is non vanishing only for the representations $S$ and $T$ ($A$).
$\mathcal{N}_{\ell}^{R}$ is a factor which ensures that the two-point function
of $J^{a}_{\mu_{1}\dots\mu_{\ell}}(x)$ is correctly normalized. The linear
combination in (2.27) make sure that the operators are in a diagonal basis
with respect to the dilatation operator, hence they are primary operators of
the dCFT. In the spin $\ell=0$ case, there are two operators

$\phi^{2}(x)=\mathcal{N}_{0}^{S}\phi_{i}\phi_{i}(x),\quad
T_{ij}(x)=\frac{1}{2}\mathcal{N}_{0}^{T}\left(\phi_{i}\phi_{j}-\frac{\delta_{ij}}{N}\phi_{k}\phi_{k}\right)(x).$
(2.28)

Their anomalous dimensions already appear at first order in $\epsilon$ and
they are

$\gamma_{S,0}=\frac{N+2}{N+8}\epsilon+O(\epsilon^{2}),\quad\gamma_{T,0}=\frac{2}{N+8}\epsilon+O(\epsilon^{2}).$
(2.29)

On the other hand, in the spin $\ell\geq 1$ case one can show (see e.g. [86])
that all the anomalous dimensions vanish at first order in $\epsilon$, and
that they are non zero only from the second order onwards

$\gamma_{R,\ell}=0+O(\epsilon^{2}),\quad\forall\,\ell\geq 1.$ (2.30)

This will be the crucial point of this work since the bulk anomalous
dimensions are precisely the data we need to compute the discontinuity of the
correlator and to reconstruct it through the dispersion relation, which we now
review.

##  3 Lorentzian inversion formulae and dispersion relation

A powerful way to study homogeneous CFTs analytically is the Lorentzian
inversion formula derived in [6]. It is an integral formula which reconstructs
the OPE data of any CFT from the double discontinuity of four-point functions.
In favorable situations, the double discontinuity receives contributions only
from very few operators and the inversion formula can extract from it an
infinite amount of CFT data [87, 11] .
For the defect case, there are two analogous formulae. A defect inversion
formula was derived in [57] and allows to extract the defect channel CFT data
from a single discontinuity

$\begin{split}b(\hat{\Delta},s)=\int_{0}^{1}\frac{dz}{2z}z^{-\frac{\hat{\tau}}{2}}\int_{1}^{\frac{1}{z}}\frac{d\bar{z}}{2\pi
i}&(1-z\bar{z})(\bar{z}-z)\bar{z}^{-\frac{\hat{\Delta}+s}{2}-2}\
{}_{2}F_{1}\left(s+1,2-\frac{q}{2},\frac{q}{2}+s,\frac{z}{\bar{z}}\right)\times\\\
&\times{}_{2}F_{1}\left(1-\hat{\Delta},1-\frac{p}{2},1+\frac{p}{2}-\hat{\Delta},z\bar{z}\right)\text{Disc}\,F(z,\bar{z}).\end{split}$
(3.1)

The coefficient function $b(\hat{\Delta},s)$ has simple poles for
$\hat{\Delta}$ equal to the dimensions of exchanged operators and residues
given by the defect OPE coefficients $b^{2}_{\hat{\Delta},s}$. Therefore
$b(\hat{\Delta},s)$ contains all the CFT data of the exchanged defect
operators. The crucial ingredient of this inversion formula is the
discontinuity

$\text{Disc}F(z,\bar{z})=F(z,\bar{z}+i\epsilon)-F(z,\bar{z}-i\epsilon),$ (3.2)

where $F(z,\bar{z}+i\epsilon)$ and $F(z,\bar{z}-i\epsilon)$ indicate that
$\bar{z}$ should be taken above or below the branch cut at $\bar{z}=1$,
leaving $z$ fixed. The defect inversion formula was derived in [57] through a
contour deformation argument, which is justified only if the integrand
vanishes sufficiently fast for large $w$, or equivalently for $w\to 0$ since
the correlator is symmetric under $w\leftrightarrow\frac{1}{w}$. More
precisely, this implies that the formula (3.1) is valid for transverse spin
$s>s_{*}$ if

$F(r,w)\sim w^{-s_{*}},\quad\quad w\to 0.$ (3.3)

This means that in general the inversion formula may miss contributions to the
CFT data from low spin operators.

There is also a bulk inversion formula, which allows to extract the bulk OPE
data from a double discontinuity [58]. The formula reads 777Compared to [58],
we rewrite the bulk inversion formula for
$\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}F(z,\bar{z})$
instead of $F(z,\bar{z})$ in order to use a more conventional definition of
double discontinuity.

$\begin{split}&c(\Delta,\ell)=c^{t}(\Delta,\ell)+(-1)^{\ell}c^{u}(\Delta,\ell),\\\
&c^{t}(\Delta,\ell)=\frac{\kappa_{\Delta+\ell}}{2}\int_{0}^{1}d^{2}z\
\mu(z,\bar{z})\
f_{\ell+d-1,\Delta-d+1}(z,\bar{z})\text{dDisc}\left(\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}F(z,\bar{z})\right),\end{split}$
(3.4)

with

$\begin{split}&\kappa_{\Delta+\ell}=\frac{\Gamma(\frac{\Delta+\ell}{2})^{4}}{2\pi^{2}\Gamma(\Delta+\ell)\Gamma(\Delta+\ell-1)},\\\
&\mu(z,\bar{z})=\frac{|z-\bar{z}|^{d-p-2}|1-z\bar{z}|^{p}}{(1-z)^{d}(1-\bar{z})^{d}},\end{split}$
(3.5)

and where $f_{\Delta,\ell}(z,\bar{z})$ are the bulk blocks. The $u$-channel
term is the same with the two external bulk operators exchanged. In this case,
the coefficient function $c(\Delta,\ell)$ has poles corresponding to the
dimensions of the operators that are exchanged in the bulk OPE and
corresponding residues given by the product of bulk three-point functions and
one-point functions, $\lambda_{\phi\phi\mathcal{O}}a_{\mathcal{O}}$. The input
of the formula is the double discontinuity defined by

$\text{dDisc}F(z,\bar{z})=F(z,\bar{z})-\frac{1}{2}F^{\circlearrowleft}(z,\bar{z})-\frac{1}{2}F^{\circlearrowright}(z,\bar{z}),$
(3.6)

where this time the functions $F^{\circlearrowleft}(z,\bar{z})$ and
$F^{\circlearrowright}(z,\bar{z})$ are obtained by taking the analytic
continuation around the point $\bar{z}=0$, leaving $z$ fixed. Just like the
defect inversion formula, the bulk inversion formula might fail for low spins
[58]. More precisely, the formula is valid for spins $\ell>\ell_{*}$ where

$\left(\frac{(w-r)(1-wr)}{rw}\right)^{\Delta_{\phi}}F(r,w)\lesssim
w^{1-\ell_{*}},\quad\quad w\rightarrow 0.$ (3.7)

The two Lorentzian inversion formulae allow to extract the defect CFT data of
the theory from certain discontinuities of the two-point functions. Therefore,
these discontinuities contain all the information that is necessary to
reconstruct the full correlator. This is made explicit in the dispersion
relation, a formula that computes the full correlator directly from a
discontinuity [8]. In the case of defect CFTs, the dispersion relation reads
[60, 59]

$F(r,w)=\int_{0}^{r}\frac{dw^{\prime}}{2\pi
i}\left(\frac{1}{w^{\prime}-w}+\frac{1}{w^{\prime}-\frac{1}{w}}-\frac{1}{w^{\prime}}\right)\text{Disc}F(r,w^{\prime}),$
(3.8)

where $\text{Disc }F(r,w)$ is the discontinuity through the cut running from
$w=0$ to $w=r$. From the definition of the variable $w$, we see that
$\text{Disc}F(r,w)=-\text{Disc}F(z,\bar{z})$ as defined in (3.2). Just like
the defect inversion formula (3.1), this dispersion relation is derived from
Cauchy’s theorem by deforming the contour around the singularities and
dropping the contributions at infinity. This contour deformation argument
misses terms that are given by low spin conformal blocks, which give
contributions at infinity. If one knows the behaviour of the correlator for
$w\to 0$ (or equivalently for $|w|\to\infty$) (3.3), one can take into account
these terms by introducing a prefactor in front of the correlator

$\tilde{F}(r,w)=\left(\frac{r}{(w-r)(\frac{1}{w}-r)}\right)^{s_{*}+1}F(r,w).$
(3.9)

By construction, $\tilde{F}(r,w)$ goes like $w^{-1}$ at large $w$ and
therefore we can safely ignore the contribution at infinity and reconstruct
this rescaled correlator from (3.8). From the point of view of the original
correlator, this implies an improved dispersion relation

$\frac{F(r,w)}{(w-r)^{s_{*}+1}(\frac{1}{w}-r)^{s_{*}+1}}=\int_{0}^{r}\frac{dw^{\prime}}{2\pi
i}\left(\frac{1}{w^{\prime}-w}+\frac{1}{w^{\prime}-\frac{1}{w}}-\frac{1}{w^{\prime}}\right)\text{Disc}\left[\frac{F(r,w^{\prime})}{(w^{\prime}-r)^{s_{*}+1}(\frac{1}{w^{\prime}}-r)^{s_{*}+1}}\right].$
(3.10)

In the following, we would like to apply the formulae introduced in this
section to the $\epsilon-$expansion of correlators in the critical $O(N)$
model. We start by introducing the leading-order correlator.

##  4 Tree level

At leading order in the $\epsilon$-expansion, the correlator contains two
terms: the free correlator without the defect which contributes to $F_{1}$ in
(2.13) and the square of the one-point function (2.10) which gives the leading
contribution to $F_{2}$ 888The defect coupling at the fixed point is not small
so the one-point function of local operators are no suppressed in the
$\epsilon$-expansion.. In particular, the perturbative expansion of $a_{\phi}$
in (2.10) reads [67, 68]

$a^{2}_{\phi}=\frac{N+8}{4}+\epsilon\frac{\left(N^{2}-3N+(N+8)^{2}\log(4)-22\right)}{8(N+8)}+\mathcal{O}(\epsilon^{2}),$
(4.1)

and the leading-order contribution to the two-point function is

$\displaystyle F_{1}^{(0)}(r,w)$
$\displaystyle=\frac{rw}{(r-w)(rw-1)},\quad\quad
F_{2}^{(0)}(r,w)={a^{2}_{\phi}}^{(0)}.$ (4.2)

Each of these two terms has a simple interpretation in one of the two
channels. The free correlator $F_{1}^{(0)}$ corresponds to the exchange of the
identity operator in the bulk channel, while the squared one-point function is
associated to the exchange of the defect identity in the defect channel. On
the other hand, as usual, to reproduce the identity in a given channel, an
infinite tower of operators is needed in the crossed one. Let us review the
CFT data of these exchanged operators.

Using the linear combinations in (A.4) one can rewrite (4.2) in terms of
$F_{S}$ and $F_{T}$ introduced in (2.20)

$\begin{split}&F^{(0)}_{S}(r,w)=\frac{{a^{2}_{\phi}}^{(0)}}{N}+\frac{rw}{(r-w)(rw-1)},\\\
&F^{(0)}_{T}(r,w)={a^{2}_{\phi}}^{(0)}.\end{split}$ (4.3)

From this expression we notice the obvious fact that the bulk identity
contributes only to the singlet exchange. On the other hand, the constant term
${a^{2}_{\phi}}^{(0)}$ can be reproduced by the exchange of two infinite
towers of twist-two spin-$\ell$ operators of the schematic form 999By this
symbolic notation we only want to indicate the number of derivatives
associated to a given twist-two primary operator.

$\displaystyle[\phi^{2}]_{S,0,\ell}=\phi^{i}\partial_{\mu_{1}}...\partial_{\mu_{\ell}}\phi_{i},\quad\quad[\phi^{i}\phi^{j}]_{T,0,\ell}=\phi^{(i}\partial_{\mu_{1}}...\partial_{\mu_{\ell}}\phi^{j)}-\textup{trace}.$
(4.4)

Their CFT data can be extracted simply by comparing (4.3) with the block
expansion (2.23) or from the Lorentzian inversion formula (3.4) using 101010As
we mentioned, the Lorentzian inversion formula does not work for low spins and
in this case it does not reproduce the contribution of the bulk identity.
Nevertheless, it still reproduces correctly all the defect CFT data of the
twist-two operators.

$\begin{split}&\text{dDisc}\left(\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}F^{(0)}_{S}(z,\bar{z})\right)=2\frac{{a^{2}_{\phi}}^{(0)}}{N}\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}},\\\
&\text{dDisc}\left(\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}F^{(0)}_{T}(z,\bar{z})\right)=2{a^{2}_{\phi}}^{(0)}\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}.\end{split}$
(4.5)

The result is 111111The twist-two operators have the same dimensions in both
channel at tree level, however they are distinct operators and have different
anomalous dimensions and OPE coefficients. [58]

$\begin{split}&\Delta^{(0)}_{S,0,\ell}=\Delta^{(0)}_{T,0,\ell}=2\Delta_{\phi}+\ell,\\\
&a\lambda^{(0)}_{T,k,\ell}=a\lambda^{(0)}_{S,k,\ell}\ N=\delta_{k,0}\
{a^{2}_{\phi}}^{(0)}\frac{2^{-\ell}\Gamma\left(\frac{\ell+1}{2}\right)^{3}}{\pi\Gamma\left(\frac{\ell}{2}+1\right)\Gamma\left(\ell+\frac{1}{2}\right)}.\end{split}$
(4.6)

where the spin $\ell$ is even. Let us comment on our labels for the operators,
which will remain the same also at order $\epsilon$. We use the index $k$ to
label the classical twist $\tau^{(0)}=\Delta^{(0)}-\ell$, specifically
$k=\frac{\tau^{(0)}}{2}-\Delta_{\phi}$. In particular, at this order, only two
families of operators enter in the bulk OPE: the identity and the twist-two
operators ($k=0$). Furthermore, for the specific case $k=0$ we know that there
is a single primary operator for a given spin $\ell$. This is no longer true
for higher values of $k$, where degeneracies may appear. Therefore, our
notation will not be able to distinguish among degenerate operator, but since
also our observable is not able to make this distinction we find this notation
convenient.

Moving on to the defect channel, we have

$\begin{split}&\hat{F}^{(0)}_{S}(r,w)={a^{2}_{\phi}}^{(0)}+\frac{rw}{(r-w)(rw-1)},\\\
&\hat{F}^{(0)}_{V}(r,w)=\frac{rw}{(r-w)(rw-1)},\end{split}$ (4.7)

where we notice again that the defect identity only enters the singlet
channel, while the bulk identity is reproduced by two infinite towers of
defect operators of the schematic form

$\displaystyle[\phi]_{S,0,s}=\partial_{\perp}^{s}\hat{\phi},\quad\quad[\phi^{\hat{i}}]_{V,0,s}$
$\displaystyle=\partial_{\perp}^{s}\hat{\phi}^{\hat{i}}.$ (4.8)

These operators have transverse spin $s$ and transverse twist
$\hat{\tau}=\hat{\Delta}-s=1$. As before one can compare (4.7) with the block
expansion (2.26) or use the inversion formula (3.1) with 121212Also in this
case the Lorentzian inversion formula fails to reproduce the contribution of
the identity.

$\begin{split}&\text{Disc}\hat{F}^{(0)}_{S}(r,w)=2\pi
i\Big{(}\frac{rw}{1-rw}\Big{)}\delta(r-w),\\\
&\text{Disc}\hat{F}^{(0)}_{V}(r,w)=2\pi
i\Big{(}\frac{rw}{1-rw}\Big{)}\delta(r-w).\\\ \end{split}$ (4.9)

The result is [57, 56]

$\begin{split}&\hat{\Delta}_{S,0,s}^{(0)}=\hat{\Delta}_{V,0,s}^{(0)}=1+s,\\\
&\hat{b}^{2(0)}_{S,m,s}=\hat{b}^{2(0)}_{V,m,s}=\delta_{m,0}.\end{split}$
(4.10)

As in the bulk case we use an additional label $m$ for the classical
transverse twist, $m=\frac{\hat{\tau}^{(0)}-1}{2}$. In this case, only twist-
one operators ($m=0$) appear in the OPE, as expected from the equation of
motion of the bulk field [57].

The aim of this work is to find the next-to-leading order corrections to these
results and this is the topic of the next section.

##  5 One loop

The idea of the defect analytic bootstrap is to use information from the bulk
theory to compute correlators in the presence of the defect. In particular,
the discontinuity relevant to the dispersion relation (3.8) is controlled by
the bulk block expansion, hence we analyse which operators appear in the
perturbative expansion of (2.23). We consider the following perturbative
expansion of the bulk channel CFT data

$\begin{split}\Delta&=\Delta^{(0)}+\epsilon\,\gamma^{(1)}+\mathcal{O}(\epsilon^{2}),\\\
\lambda_{\phi\phi\Delta}a_{\Delta}&=a\lambda^{(0)}+\epsilon\,a\lambda^{(1)}+\mathcal{O}(\epsilon^{2}).\\\
\end{split}$ (5.1)

More precisely, operators that appeared at tree level, namely twist-two
operators, enter in the one-loop OPE with their anomalous dimensions

$\Delta_{S/T,0,\ell}=2\Delta_{\phi}+\ell+\epsilon\gamma^{(1)}_{S/T,0,\ell}+\mathcal{O}(\epsilon^{2}),$
(5.2)

which are known from previous work on the $O(N)$ model without defects (see
for example [83] and references therein)

$\displaystyle\gamma^{(1)}_{S,0,\ell}=\frac{N+2}{N+8}\delta_{0,\ell},$
$\displaystyle\gamma^{(1)}_{T,0,\ell}=\frac{2}{N+8}\delta_{0,\ell}.$ (5.3)

Notice the crucial fact that only the spin-one operator of each representation
has a non-vanishing anomalous dimension at one loop. This will be extremely
important for the computation of the discontinuity.

The second contribution we need to consider originates from the anomalous
dimension of the external operator $\phi_{i}$

$\Delta_{\phi}=1-\frac{\epsilon}{2}+\epsilon\gamma^{(1)}_{\phi}+\mathcal{O}(\epsilon^{2}),$
(5.4)

which gives a correction to the bulk identity contribution

$\begin{split}\textstyle\left(\frac{rw}{(r-w)(rw-1)}\right)^{\Delta_{\phi}}=\frac{rw}{(r-w)(rw-1)}+\epsilon\left(\gamma^{(1)}_{\phi}-\frac{1}{2}\right)\frac{rw}{(1-rw)(w-r)}\log\Big{(}\frac{rw}{(1-rw)(w-r)}\Big{)}+\mathcal{O}(\epsilon^{2}).\end{split}$
(5.5)

Specifically, the anomalous dimension $\gamma^{(1)}_{\phi}$ is well-known to
be vanishing at one loop, i.e. $\gamma^{(1)}_{\phi}=0$, but clearly this does
not kill the contribution $\eqref{bulkidentity}$.

In principle, we can also have operators with higher twist, which appear in
the OPE with their classical dimensions

$\begin{split}\Delta_{S/T,k,\ell}=2\Delta_{\phi}+2k+\ell+\mathcal{O}(\epsilon),\
\ \ \ k>0.\end{split}$ (5.6)

In particular, we expect only operators of twist up to $4$, since the bulk OPE
coefficients of higher-twist operators are of order $\epsilon^{2}$ [9, 10].
Nevertheless, while there is a single family of operators with twist two, i.e.
a single primary operator for each spin, starting from twist four there can be
degenaracies which cannot be lifted by studying a single correlator. Luckily,
these operators do not contribute to the discontinuity in (3.8) so this is not
an issue to reconstruct the full correlator. We will comment further on this
issue in Section 5.3.

All in all, the one-loop bulk block expansion for the singlet reads

$\begin{split}F^{(1)}_{S}(r,w)&=-\frac{rw}{2(1-rw)(w-r)}\log\Big{(}\frac{rw}{(1-rw)(w-r)}\Big{)}+\\\
&+\frac{rw}{2(1-rw)}a\lambda^{(0)}_{S,0,0}\gamma^{(1)}_{S,0,0}\left(\tilde{f}_{2,0}(r,w)\log(w-r)+\partial_{\Delta}\tilde{f}_{2,0}(r,w)\right)+\\\
&+\sum_{k=0}^{1}\sum_{\ell}\
\frac{rw}{1-rw}a\lambda^{(1)}_{S,k,\ell}(w-r)^{k}\tilde{f}_{2+2k+\ell,\ell}(r,w),\end{split}$
(5.7)

where we defined
$\tilde{f}_{\Delta,\ell}(r,w)=(w-r)^{\frac{-\Delta+\ell}{2}}f_{\Delta,\ell}$,
with $f_{\Delta,\ell}$ given in (A.10). For the symmetric traceless part we
have

$\begin{split}F^{(1)}_{T}(r,w)&=\frac{rw}{2(1-rw)}a\lambda^{(0)}_{T,0,0}\gamma^{(1)}_{T,0,0}\left(\tilde{f}_{2,0}(r,w)\log(w-r)+\partial_{\Delta}\tilde{f}_{2,0}(r,w)\right)+\\\
&+\sum_{k=0}^{1}\sum_{\ell}\
\frac{rw}{1-rw}a\lambda^{(1)}_{T,k,\ell}(w-r)^{k}\tilde{f}_{2+2k+\ell,\ell}(r,w).\end{split}$
(5.8)

We can consider an analogous expansion for the defect channel CFT data

$\begin{split}\hat{\Delta}&=\hat{\Delta}^{(0)}+\epsilon\,\hat{\gamma}^{(1)}+\mathcal{O}(\epsilon^{2}),\\\
\hat{b}&=\hat{b}^{(0)}+\epsilon\,\hat{b}^{(1)}+\mathcal{O}(\epsilon^{2}).\end{split}$
(5.9)

As before, operators with transverse twist one, which already appeared at tree
level, enter the block expansion with their anomalous dimension

$\hat{\Delta}_{S/V,0,s}=1+s+\epsilon\,\hat{\gamma}^{(1)}_{S/T,0,s}+\mathcal{O}(\epsilon^{2}),$
(5.10)

while higher-twist operators contribute with their classical dimensions

$\begin{split}\hat{\Delta}_{S/V,m,s}=1+2m+s+\mathcal{O}(\epsilon),\ \ \ \
m>0.\end{split}$ (5.11)

In particular, at one loop we expect only operators of transverse twist one,
i.e. $m=0$, since higher-twist defect operators have bulk-to-defect couplings
$\hat{b}$ of order $\epsilon$, hence their squared coefficients are at least
of order $\epsilon^{2}$. We will see later on that this expectation is
confirmed by the inversion formula. The defect block expansions at one loop
read

$\begin{split}&\hat{F}^{(1)}_{S}(r,w)={a^{2}_{\phi}}^{(1)}+\sum_{s}\
\hat{b}^{2(1)}_{S,0,s}\hat{f}_{1+s,s}(r,w)+\hat{b}^{2(0)}_{S,0,s}\hat{\gamma}^{(1)}_{S,0,s}\partial_{\hat{\Delta}}\hat{f}_{1+s,s}(r,w),\\\
&\hat{F}^{(1)}_{V}(r,w)=\sum_{s}\
\hat{b}^{2(1)}_{V,0,s}\hat{f}_{1+s,s}(r,w)+\hat{b}^{2(0)}_{V,0,s}\hat{\gamma}^{(1)}_{V,0,s}\partial_{\hat{\Delta}}\hat{f}_{1+s,s}(r,w).\end{split}$
(5.12)

Before showing results for the defect CFT data let us compute the full one-
loop correlator using the dispersion relation.

###  5.1 The full result

We can use the dispersion relation (3.8) to compute the full correlator at one
loop provided the correlator is sufficiently well-behaved for $w\to 0$ as
discussed in (3.3). In particular, we assume that after subtracting the defect
identity (which is just the one-loop correction to the squared one-point
function $a_{\phi}^{2}$), the rest of the correlator vanishes for $w\to 0$,
i.e. that $s_{*}<0$ in (3.3). We checked that our assumption is correct by
comparing the non-trivial part of the full correlator obtained from the
dispersion relation (5.15) with the numerical integration of the result from
Feynman diagrams in Appendix C.2. We found perfect agreement between the two
results.

The main ingredient of the dispersion relation is the discontinuity (3.2). The
discontinuity can be computed term by term in the bulk-channel block
expansion. It is convenient to consider the singlet and traceless contribution
separately. Starting from the singlet representation, we see from (5.7) that
the discontinuity at first order in the $\epsilon$-expansion is given only by
the logarithmic terms, because the blocks $\tilde{f}_{\Delta,\ell}(r,w)$ and
their derivatives, evaluated at the tree-level dimensions (4.6), are regular
at $w=r$. Crucially, these terms are proportional to tree-level data and one-
loop anomalous dimensions. Notice that (5.3), combined with (5.7), implies
that the one-loop discontinuity of the singlet is just given by two terms: one
is the correction to the bulk identity from the engineering dimension of the
external field and the other one is proportional to a single bulk block. Using

$\text{Disc}(\log(w-r))=2\pi i,$ (5.13)

and the explicit form of the bulk blocks as a series expansion (A.10) we find

$\begin{split}&\text{Disc}F^{(1)}_{S}(r,w)=-\pi i\frac{(rw)}{(r-w)(rw-1)}+2\pi
i\Big{(}\frac{rw}{2(1-rw)}\Big{)}a\lambda^{(0)}_{S,0,0}\gamma^{(1)}_{S,0,0}\tilde{f}_{2,0}(r,w)=\\\
&=-\textstyle\pi
i\frac{(rw)}{(r-w)(rw-1)}+\frac{N+2}{8N}\frac{4\pi\sqrt{rw}\left(F\left(\sin^{-1}\left(\sqrt{r}\sqrt{\frac{r-w}{1-rw}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)-F\left(\sin^{-1}\left(\frac{\sqrt{\frac{r-w}{1-rw}}}{\sqrt{r}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)\right)}{r-w},\end{split}$
(5.14)

where $F(x,k)$ is the incomplete elliptic integral of the first kind. We
stress that this contribution is given by a single operator in the bulk
expansion, i.e. the operator of twist two and spin zero $\phi^{2}$, whose
anomalous dimension (5.3) is our only input. The complicated form of the
function (5.14) is due to the bulk conformal block $\tilde{f}_{2,0}$ which is
a particular case of the complicated expression in (A.10).

The first term in (5.14) trivially reproduces the correction to the bulk
identity (5.5), as we expected. The second contribution is very complicated
and we could not solve the corresponding integral in the dispersion relation
in terms of simple functions. However, we can find the result as a series
expansion or as a derivative of a special function. We refer to (B) for the
explicit computation. At the end of the day we find

$\begin{split}&\textstyle F^{(1)}(r,w)_{S,\text{not
id}}=\frac{N+2}{8N}\sum_{m=0}^{\infty}\frac{2^{1-m}rw(1-r^{2})^{m}G_{4,4}^{4,2}\left(\frac{4w}{wr^{2}-\left(w^{2}+1\right)r+w}|\begin{array}[]{c}0,0,\frac{m}{2},\frac{m+1}{2}\\\
-\frac{1}{2},0,m,m\\\
\end{array}\right)}{(m!)\left(r^{2}(-w)+rw^{2}+r-w\right)}=\\\
&=\textstyle\frac{N+2}{8N}\frac{\partial}{\partial
t}\left(\left(\frac{(r-w)(rw-1)}{(1+r)^{2}w}\right)^{t}\frac{4r}{(1+r)^{2}(2t+1)}\left.F^{112}_{101}\left(\begin{array}[]{c}1+t:\frac{1}{2};\frac{1}{2}+t,1;\\\
\frac{3}{2}+t:-;1+t;\\\ \end{array}\
\left(\frac{1-r}{1+r}\right)^{2},\frac{(r-w)(rw-1)}{(1+r)^{2}w}\right)\right)\right|_{t=0}\\\
&\equiv\frac{N+2}{8N}I(r,w),\end{split}$ (5.15)

where the result is expressed in terms of an infinite sum of Meijer
G-functions $G_{4,4}^{4,2}$ or a derivative of a Kampè de Fèriet function
$F_{101}^{112}$ [88], whose definitions are given in Appendix B. In the last
line of (5.15) we defined the function $I(r,w)$ for later convenience. Putting
all together and adding back the contribution of the defect identity, the
singlet contribution reads

$\begin{split}F^{(1)}_{S}(r,w)&=\frac{{a^{2}_{\phi}}^{(1)}}{N}-\Big{(}\frac{rw}{2(1-rw)(w-r)}\Big{)}\log\Big{(}\frac{rw}{(1-rw)(w-r)}\Big{)}+\frac{N+2}{8N}I(r,w).\end{split}$
(5.16)

We can follow the same argument for the symmetric traceless part, the only
difference is that the bulk identity contribution is absent and the overall
factor changes because of the OPE data (4.6) and (5.3). The discontinuity
reads

$\begin{split}&\text{Disc}F^{(1)}_{T}(r,w)=2\pi
i\Big{(}\frac{rw}{2(1-rw)}\Big{)}a\lambda^{(0)}_{T,0,0}\gamma^{(1)}_{T,0,0}\tilde{f}_{2,0}(r,w)=\\\
&=\frac{\pi\sqrt{rw}\left(F\left(\sin^{-1}\left(\sqrt{r}\sqrt{\frac{r-w}{1-rw}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)-F\left(\sin^{-1}\left(\frac{\sqrt{\frac{r-w}{1-rw}}}{\sqrt{r}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)\right)}{r-w}.\end{split}$
(5.17)

At the end of the day we find

$\begin{split}F^{(1)}_{T}(r,w)&={a^{2}_{\phi}}^{(1)}+\frac{1}{4}I(r,w).\end{split}$
(5.18)

Using the combinations in (A.4) we can rewrite the results in terms of the
functions $\hat{F}_{S}$ and $\hat{F}_{T}$ that are natural in the defect
channel as

$\begin{split}&\hat{F}_{S}^{(1)}(r,w)={a^{2}_{\phi}}^{(1)}-\Big{(}\frac{rw}{2(1-rw)(w-r)}\Big{)}\log\Big{(}\frac{rw}{(1-rw)(w-r)}\Big{)}+\frac{3}{8}I(r,w),\\\
&\hat{F}_{V}^{(1)}(r,w)=-\Big{(}\frac{rw}{2(1-rw)(w-r)}\Big{)}\log\Big{(}\frac{rw}{(1-rw)(w-r)}\Big{)}+\frac{1}{8}I(r,w).\end{split}$
(5.19)

We now move to the computation of the defect CFT data which appear in the two
channels.

###  5.2 Defect channel data

In principle, we could extract the defect CFT data in the defect channel by
plugging the discontinuity in the defect inversion formula (3.1), under the
same assumptions as in the case of the dispersion relation, namely that we are
not missing any low spin contribution except for the defect identity. However,
as usual, solving the integral to find the full coefficient function
$b(\hat{\Delta},s)$ is very hard. For this reason, we follow the approach of
[6], which was specialized to the case of defects in [57], and perform a small
$z$ expansion in the inversion formula and only then we compute the integral
for each term in the expansion. If we expand the defect inversion integral at
small $z$ we find

$\begin{split}&b(\hat{\Delta},s)=\int_{0}^{1}\frac{dz}{2z}z^{-\frac{\hat{\tau}}{2}}\sum_{m=0}z^{m}\sum_{k=-m}^{m}c_{m,k}(\hat{\Delta},s)B(z,\beta+2k),\\\
&B(z,\beta)=\int_{1}^{\infty}\frac{d\bar{z}}{2\pi
i}\bar{z}^{-\frac{\beta}{2}-1}\text{
Disc}\hat{F}_{S,V}(z,\bar{z}),\end{split}$ (5.20)

where $\beta=\hat{\Delta}+s=\hat{\tau}+2s$ and where $c_{m,k}(\hat{\Delta},s)$
are the coefficients obtained from the small $z$ expansion of the integrand of
(3.1). Notice that a term proportional to $z^{\alpha}$ in the small $z$
expansion reproduces the contribution of twist $2\alpha$ in the coefficient
function $b(\hat{\Delta},s)$, because the last integral is

$\int_{0}^{1}dz\
\frac{z^{-\frac{\hat{\tau}}{2}}}{2z}z^{\alpha}=-\frac{1}{\hat{\tau}-2\alpha}.$
(5.21)

We will show explicitly the computation of the coefficient only for the
leading order in the small $z$ expansion, which allows to extract the CFT data
for transverse twist equal to $1$. We will just give the results for higher
orders because the expressions become more complicated.
Starting from (5.14) and (5.17) and performing the linear combinations (A.4),
we obtain

$\begin{split}&\textstyle\text{Disc}\hat{F}_{S}^{(1)}(r,w)=-\frac{\pi
i(rw)}{(r-w)(rw-1)}+\frac{3\pi\sqrt{rw}\left(F\left(\sin^{-1}\left(\sqrt{r}\sqrt{\frac{r-w}{1-rw}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)-F\left(\sin^{-1}\left(\frac{\sqrt{\frac{r-w}{1-rw}}}{\sqrt{r}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)\right)}{2(r-w)},\\\
&\textstyle\text{Disc}\hat{F}_{V}^{(1)}(r,w)=-\frac{\pi
i(rw)}{(r-w)(rw-1)}+\frac{\pi\sqrt{rw}\left(F\left(\sin^{-1}\left(\sqrt{r}\sqrt{\frac{r-w}{1-rw}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)-F\left(\sin^{-1}\left(\frac{\sqrt{\frac{r-w}{1-rw}}}{\sqrt{r}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)\right)}{2(r-w)}.\end{split}$
(5.22)

We see that in both the singlet and the vector defect representations we have
two contributions in the discontinuity 131313Notice that in both channels the
discontinuity does not depend on $N$, therefore the only data in the defect
OPE that will depend on $N$ will be the defect identity correction, which is
missed by the inversion formula.. The first one comes from the the bulk
identity operator. This contribution was already considered in [57] for a
generic value of $\Delta_{\phi}$ and it gives

$\begin{split}B(z,\beta)_{\text{id}}=\frac{\sin(\pi\Delta_{\phi})\Gamma(1-\Delta_{\phi})\left(-\frac{\sqrt{z}}{z-1}\right)^{\Delta_{\phi}}\Gamma\left(\frac{\beta+\Delta_{\phi}}{2}\right)}{\pi\Gamma\left(\frac{1}{2}(\beta-\Delta_{\phi}+2)\right)}.\end{split}$
(5.23)

Expanding in $\epsilon$ and selecting the first order gives

$\begin{split}B(z,\beta)^{(1)}_{\text{id}}&=\frac{\sqrt{z}\left(\psi^{(0)}\left(\frac{\beta+1}{2}\right)+\log\left(-\frac{\sqrt{z}}{z-1}\right)+\gamma\right)}{2(z-1)}=\\\
&=\frac{1}{4}\sqrt{z}\left(-2\psi^{(0)}\left(\frac{1+\beta}{2}\right)-\log(z)-2\gamma\right)+O(z^{\frac{3}{2}}),\end{split}$
(5.24)

where $\psi^{(0)}(z)$ is the digamma function. This result has to be combined
with the second contribution in equation (5.22), which can be expanded as

$\begin{split}\text{Disc}\hat{F}_{S}(z,\bar{z})_{\text{not
id}}=\frac{3}{4}i\pi\sqrt{z}\left(\log(z)+\log(\bar{z})-4\log\left(\frac{2\sqrt{\bar{z}}}{\sqrt{\bar{z}}+1}\right)\right)+O(z^{\frac{3}{2}}),\\\
\text{Disc}\hat{F}_{V}(z,\bar{z})_{\text{not
id}}=\frac{1}{4}i\pi\sqrt{z}\left(\log(z)+\log(\bar{z})-4\log\left(\frac{2\sqrt{\bar{z}}}{\sqrt{\bar{z}}+1}\right)\right)+O(z^{\frac{3}{2}}).\end{split}$
(5.25)

Inserting these discontinuities into (5.20) we find

$\begin{split}&B_{S}(z,\beta)_{\text{not
id}}=\frac{3\sqrt{z}\left(2(\beta+1)-2(\beta-1)\beta\left(\psi^{(0)}\left(\frac{\beta}{2}\right)+\gamma\right)+(\beta-1)\beta\left(2H_{\frac{\beta-3}{2}}+\log(z)\right)\right)}{4(\beta-1)\beta^{2}},\\\
&B_{V}(z,\beta)_{\text{not
id}}=\frac{\sqrt{z}\left(2(\beta+1)-2(\beta-1)\beta\left(\psi^{(0)}\left(\frac{\beta}{2}\right)+\gamma\right)+(\beta-1)\beta\left(2H_{\frac{\beta-3}{2}}+\log(z)\right)\right)}{4(\beta-1)\beta^{2}}.\end{split}$
(5.26)

where $H_{z}$ are harmonic numbers. Combining both contributions (5.24) and
(5.26) in (5.20) we obtain

$\begin{split}b_{S}(\hat{\Delta},s)&\sim\frac{s-1}{(2s+1)(\hat{\tau}-1)^{2}}+\frac{2(s-1)H_{s}+3H_{s+\frac{1}{2}}}{2(2s+1)(\hat{\tau}-1)},\\\
b_{V}(\hat{\Delta},s)&\sim\frac{s}{(2s+1)(\hat{\tau}-1)^{2}}+\frac{(2s+1)\left(2sH_{s}+H_{s-\frac{1}{2}}\right)+2}{2(2s+1)^{2}(\hat{\tau}-1)}.\end{split}$
(5.27)

The appearance of double poles signals the presence of anomalous dimensions,
indeed

$b(\hat{\Delta},s)\sim\frac{\hat{b}^{(0)}+\epsilon\hat{b}^{(1)}}{\hat{\tau}^{(0)}+\epsilon\hat{\gamma}^{(1)}-1}=\frac{\hat{b}^{(1)}}{\hat{\tau}^{(0)}-1}-\frac{\hat{b}^{(0)}\hat{\gamma}^{(1)}}{(\hat{\tau}^{(0)}-1)^{2}}.$
(5.28)

Comparing with (5.27), we find the one-loop defect data

$\displaystyle\hat{\gamma}_{S,0,s}^{(1)}=\frac{1-s}{(2s+1)},$
$\displaystyle\hat{b}_{S,0,s}^{2(1)}=\frac{-2(s-1)H_{s}-3H_{s+\frac{1}{2}}}{2(2s+1)},$
$\displaystyle\hat{\gamma}_{V,0,s}^{(1)}=-\frac{s}{(2s+1)},$
$\displaystyle\hat{b}_{V,0,s}^{2(1)}=-\frac{(2s+1)\left(2sH_{s}+H_{s-\frac{1}{2}}\right)+2}{2(2s+1)^{2}}.$
(5.29)

We can perform several sanity checks on these results. First of all, we can
check that $\hat{\gamma}_{S,0,1}^{(1)}=0$, i.e. the presence of the
displacement operator in the defect OPE of the fundamental field. The
displacement operator is a protected operator, which is related to the
explicit breaking of translational symmetry and, for a line defect, it has
dimension two. Furthermore, it has orthogonal spin one and it is a singlet
under internal symmetries. The only candidate in this case is the operator
$\partial_{\perp}\hat{\phi}^{1}$ which is indeed associated to the vanishing
anomalous dimension $\hat{\gamma}_{S,0,1}^{(1)}$.

Another universal protected operator is the so-called tilt operator which
appears whenever the defect breaks part of the internal symmetry of the bulk
theory. In this case the defect breaks $O(N)$ to $O(N-1)$ and each of the
$N-1$ broken generators is associated to a tilt operator forming a vectorial
representation of the preserved subgroup. This vector has dimension one and
orthogonal spin zero, i.e. it is given precisely by $\hat{\phi}^{\hat{i}}$,
the $N-1$ scalars that are not involved in the construction of the defect.
Correspondingly, we find $\hat{\gamma}_{V,0,0}^{(1)}=0$.

For other defect operators some results are already avaliable in the
literature. The anomalous dimensions for the spin-zero singlet, i.e.
$\hat{\phi}^{1}$ and the spin-one vector operators, i.e.
$\partial_{\perp}\hat{\phi}^{\hat{i}}$, have been computed in equations (3.19)
and (3.52) of [68] and we find perfect agreement. More generally, the leading-
twist contribution to the defect anomalous dimensions from the inversion of a
single bulk scalar operator of arbitrary dimension $\Delta$ was computed in
(3.53) of [57]. The perturbative expansion of that result perfectly matches
ours. Our results for the bulk-to-defect couplings are completely new.

As we already mentioned, the one-loop result contains information about the
anomalous dimensions of operators that already appeared at tree level.
Nevertheless, keeping more terms in the small-$z$ expansion (5.20) and (5.25)
we can also extract the bulk-to-defect couplings for higher-twist operators.
As one could have guessed from a diagrammatic argument and as we anticipated
in (5.12) it turns out that all higher-twist coefficients are zero at this
order in perturbation theory.

$\displaystyle\hat{b}_{S,m,s}^{2(1)}=\hat{b}_{V,m,s}^{2(1)}=0,\quad m>0.$
(5.30)

Therefore the only non zero OPE data in the defect channel are (5.29) and the
defect identity at one loop (4.1).

###  5.3 Bulk channel data

Finally, we can extract the bulk data using the bulk inversion formula (3.4).
The bulk anomalous dimensions are well-known and reproducing them is just a
consistency check for the validity of our procedure. On the other hand, the
product $a\lambda^{(1)}$ depends on the defect through the one-point function
$a$ and therefore we can provide new predictions for all the operators not
afflicted by perturbative degeneracies (in this case all the twist-two
operators and the first two operator in the twist-four family).
In order to use the formula (3.4), we need to analyse the behaviour of
$\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}F(z,\bar{z})$
for $w\to 0$ according to (3.7) 141414Notice that for the bulk inversion we
need to consider an extra factor in front of the correlator $F(r,w)$. For this
reason the behaviour at small $w$ is different from the one discussed in the
defect inversion section.. In our case, we can see that $\ell_{*}=2$ by
expanding the results of the dispersion relation (5.16) and (5.18) around
$w=0$ and comparing with (3.7), therefore the inversion formula will work for
$\ell>2$.
Since the integral in (3.4) is too hard, we follow the same strategy that we
used in the case of the defect inversion. We expand the integrand around $z=1$
and compute the coefficient term by term in the expansion. Namely we use [58]

$\begin{split}&c^{t}(\Delta,\ell)=\int_{0}^{1}\frac{dz}{2(1-z)}(1-z)^{\frac{\ell-\Delta}{2}}\sum_{m=0}(1-z)^{m}\sum_{k=-m}^{m}B_{m,k}(\Delta,\ell)C^{t}(z,\Delta+\ell+2k),\\\
&C^{t}(z,\beta)=\kappa_{\beta}\int_{0}^{z}\frac{d\bar{z}}{(1-\bar{z})^{2}}k_{\beta}(1-\bar{z})\text{dDisc}\left(\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}F(z,\bar{z})\right),\\\
&k_{\beta}(z)=z^{\frac{\beta}{2}}{}_{2}F_{1}(\frac{\beta}{2},\frac{\beta}{2},\beta,z),\end{split}$
(5.31)

where $B_{m,k}(\Delta,\ell)$ are coefficients that can be fixed by expanding
the kernel in (3.4) and comparing with (5.31), namely

$\begin{split}&(1-z)(1-\bar{z})^{2}(1-z)^{\frac{\Delta-l}{2}}\mu(z,\bar{z})f^{HS}_{d+l-1,-d+\Delta+1}=\\\
&=\sum_{m}(1-z)^{m}\sum_{k=-m}^{m}\frac{\kappa_{\Delta+2k+l}}{\kappa_{\Delta+l}}B_{m,k}(\Delta,\ell)k_{\Delta+\ell+2k}(1-\bar{z}).\end{split}$
(5.32)

Just like in the defect case, a given term in the series expansion around
$z=1$ (5.31) reproduces the contribution of a given twist to the coefficient.
We can compute the double discontinuity at order $\epsilon$ by taking the
expressions for the full correlator (5.16) and (5.18), multiplying by
$\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}$ and expanding in $z=1$. Every term
in the expansion of the non trivial part of the correlator
$\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}I(r,w)$ contains only integer powers
of $\bar{z}$. Therefore, we see that this part of the correlator does not
contribute to the double discontinuity. The same applies for the contribution
from the correction to the bulk identity. In other words, the only
contributions to the double discontinuity of both representations at one loop
are the terms proportional to the defect identity,
${a^{2}_{\phi}}^{(1)}\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}$. However, the
one-loop coefficient (5.31) will also receive a contribution from the tree-
level double discontinuity because it contains factors that depend on
$\Delta_{\phi}$ and $d$, which will give terms of order $\epsilon$ when
combined with the order zero double discontinuity (4.5). In particular both
$C^{t}(z,\beta)$ and $B_{m,k}(\Delta,\ell)$ depend on $\epsilon$ through
$\Delta_{\phi}$ and $d$. All in all, we need to consider

$\begin{split}&\textstyle\text{dDisc}\left(\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}F_{S}(r,w)\right)=\left(\frac{{a^{2}_{\phi}}^{(0)}}{N}+\epsilon\frac{{a^{2}_{\phi}}^{(1)}}{N}\right)2\sin^{2}\left(\frac{\pi\Delta_{\phi}}{2}\right)\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}},\\\
&\textstyle\text{dDisc}\left(\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}F_{T}(r,w)\right)=\left({a^{2}_{\phi}}^{(0)}+\epsilon{a^{2}_{\phi}}^{(1)}\right)2\sin^{2}\left(\frac{\pi\Delta_{\phi}}{2}\right)\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}.\end{split}$
(5.33)

We computed the first few contributions to $C^{t}(z,\beta)$ in the $z=1$
expansion in the appendix (B.8), for both representations. Plugging those
expressions in (5.31), using (4.1) and expanding at first order in $\epsilon$,
we can find the one-loop coefficients $c_{S/T}^{t}(\Delta,\ell)$. Since the
external operators are identical, the u-channel contribution
$c_{S/T}^{u}(\Delta,\ell)$ is the same. At the end of the day, we can extract
the one-loop bulk OPE data for both representations

$\begin{split}&a\lambda^{(1)}_{T,0,\ell}=-\frac{2^{-\ell-5}\Gamma\left(\frac{\ell}{2}+\frac{1}{2}\right)^{3}}{\pi\Gamma\left(\frac{\ell}{2}+1\right)\Gamma\left(\ell+\frac{1}{2}\right)}\Big{(}-32{a^{2}_{\phi}}^{(0)}H_{\frac{\ell}{2}-\frac{1}{2}}+35{a^{2}_{\phi}}^{(0)}H_{\ell-\frac{1}{2}}+19{a^{2}_{\phi}}^{(0)}\psi^{(0)}(\ell)+\\\
&-38{a^{2}_{\phi}}^{(0)}\psi^{(0)}(2\ell)-19\gamma{a^{2}_{\phi}}^{(0)}+6{a^{2}_{\phi}}^{(0)}\log(2)+32{a^{2}_{\phi}}^{(1)}\Big{)},\\\
&a\lambda^{(1)}_{T,1,\ell}=-\frac{{a^{2}_{\phi}}^{(0)}2^{-\ell-3}\Gamma\left(\frac{\ell+1}{2}\right)\Gamma\left(\frac{\ell+3}{2}\right)^{2}}{\pi\Gamma\left(\frac{\ell}{2}+2\right)\Gamma\left(\ell+\frac{3}{2}\right)},\\\
&a\lambda^{(1)}_{S,0,\ell}=\frac{1}{N}\lambda^{(1)}_{T,0,\ell},\\\
&a\lambda^{(1)}_{S,1,\ell}=\frac{1}{N}\lambda^{(1)}_{T,1,\ell}.\end{split}$
(5.34)

The anomalous dimensions are all zero for $\ell>0$, as we already knew. We
also computed $C^{t}(z,\beta)$ and (5.31) to higher order in the $z=1$
expansion to extract the coefficients and anomalous dimensions of higher-twist
operators and checked that they turn out to be zero. The data at low spin have
to be computed in a different way, because the inversion formula does not
converge in that case. For example we can compute the missing data by
expanding the full results (5.16) and (5.18) in series and comparing it to the
bulk OPE expansion. Although we expected the inversion formula to fail at spin
$\ell=0,2$, we find out that the only data that are missed by the inversion
formula are the coefficients of the twist-two operators with $\ell=0$ (2.21).
We obtain

$\begin{split}&a\lambda^{(1)}_{T,0,0}={a^{2}_{\phi}}^{(1)}-\frac{1}{2}-\frac{\log(2)}{2},\\\
&a\lambda^{(1)}_{S,0,0}=-\frac{-4{a^{2}_{\phi}}^{(1)}+N+N\log(2)+2+\log(4)}{4N}.\end{split}$
(5.35)

As another non-trivial check of our results at twist $4$ at low spin, we can
follow an alternative route to derive the coefficient of the operator
$\phi^{4}$, namely $a\lambda^{(1)}_{S/T,1,0}$, and compare with the result
from the inversion formula. For the sake of simplicity we set $N=1$ and
therefore consider only the singlet representation. We can take the two-point
function of the operator $\phi^{2}$ at tree level and extract the bulk CFT
data. Just like in the case of the two-point function of $\phi$, the tree-
level correlator has a contribution from the squared one-point function and
from the bulk identity with dimension $\Delta_{\phi^{2}}=2+O(\epsilon)$,
namely

$F^{(0)}(r,w)={a^{2}_{\phi^{2}}}^{(0)}+{a^{2}_{\phi}}^{(0)}\frac{rw}{(r-w)(rw-1)}+\left(\frac{rw}{(r-w)(rw-1)}\right)^{2}.$
(5.36)

A family of double-twist operators with twist $4$ enters in the bulk spectrum
at this order. In particular, we can extract the coefficient of the spin zero
double twist
$a\lambda^{(0)}_{\phi^{2}\phi^{2}\phi^{4}}=a^{(0)}_{\phi^{4}}\lambda^{(0)}_{\phi^{2}\phi^{2}\phi^{4}}$
from the bulk expansion. Since the squared three-point coefficients
$\lambda^{2\ (0)}_{\phi^{2}\phi^{2}\phi^{4}}$ are known from previous work on
the theory without defects [83], one can immediately derive an expression for
the squared one-point function $a^{2\ (0)}_{\phi^{4}}$ from
$a\lambda^{(0)}_{\phi^{2}\phi^{2}\phi^{4}}$. Multiplying this expression and
the known squared one-loop OPE coefficients $\lambda^{2\
(1)}_{\phi\phi\phi^{4}}$, one can compute the square of the $\phi^{4}$
coefficient $a^{2\ (0)}_{\phi^{4}}\lambda^{2\
(1)}_{\phi\phi\phi^{4}}=a\lambda^{2(1)}_{S,1,\ell}$, where we used the fact
that $\lambda^{2\ (0)}_{\phi\phi\phi^{4}}=0$. The results we obtain match with
what we expect from (5.34). Obviously this reasoning can be extended to any
$N$. Considering other correlators is not only useful to check results but
also to solve the mixing problem. Indeed, starting at spin $\ell=2$, the
twist-four operators suffer from degeneracies, namely there are multiple
operators that have the same classical scaling dimensions and therefore are
indistinguishable from the bootstrap perspective. For this reason the data
that we found at twist four (5.34) has to be interpreted as an average over
all degenerate twist-four operators for a given spin. In order to lift the
degeneracy, one should in principle compare the results from different
correlators.

## Acknowledgments

We would like to thank A. Gimenez-Grau for coordinating the submission of his
work with ours. We are also grateful to M. Billò, A. Bissi, G. Bliard, V.
Forini, F. Galvagno, P. Garau, M. Meineri, M. Lemos, P. van Vliet for useful
discussions. The research of LB is funded through the MIUR program for young
researchers “Rita Levi Montalcini”. The research of DB received partial
support through the STFC grant ST/S005803/1.

##  Appendix A Kinematics and conformal blocks

In this work we consider the two-point function

$\braket{\phi_{i}(x)\phi_{j}(y)}=\frac{F_{1}(z,\bar{z})\delta_{ij}+F_{2}(z,\bar{z})\delta_{i1}\delta_{j1}}{|x_{\perp}|^{\Delta_{\phi}}|y_{\perp}|^{\Delta_{\phi}}},$
(A.1)

where $z$ and $\bar{z}$ are the lightcone coordinates defined by

$\xi_{1}=\frac{(x-y)^{2}}{4|x_{\perp}||y_{\perp}|},\quad\xi_{2}=\frac{x\cdot
y}{|x_{\perp}||y_{\perp}|};\quad\xi_{1}=\frac{(1-z)(1-\bar{z})}{4\sqrt{z\bar{z}}},\quad\xi_{2}=\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}.$
(A.2)

They are related to the radial coordinates by

$z=rw,\quad\bar{z}=\frac{r}{w}.$ (A.3)

The functions $F_{1}(z,\bar{z})$ and $F_{2}(z,\bar{x})$ can be expressed as
linear combinations of $F_{S}(z,\bar{z})$, $F_{T}(z,\bar{z})$ and of
$\hat{F}_{S}(z,\bar{z})$, $\hat{F}_{T}(z,\bar{z})$ defined in (2.20) and
(2.24)

$\begin{split}&F_{1}(z,\bar{z})=F_{S}(z,\bar{z})-\frac{1}{N}F_{T}(z,\bar{z})=\hat{F}_{V}(z,\bar{z}),\\\
&F_{2}(z,\bar{z})=F_{T}(z,\bar{z})=\hat{F}_{S}(z,\bar{z})-\hat{F}_{V}(z,\bar{z}).\end{split}$
(A.4)

The functions $F_{1}(z,\bar{z})$ and $F_{2}(z,\bar{z})$ obey crossing
equations

$\begin{split}F_{1}(z,\bar{z})=\left(\frac{\sqrt{z\bar{z}}}{(1-z)(1-\bar{z})}\right)^{\Delta_{\phi}}\sum_{\Delta,\ell}\,\bigg{(}a_{\mathcal{O}_{S}}\,&\lambda_{\phi\phi\mathcal{O}_{S}}-\frac{1}{N}a_{\mathcal{O}_{T}}\,\lambda_{\phi\phi\mathcal{O}_{T}}\bigg{)}\,f_{\Delta,\ell}(z,\bar{z})=\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \
=\sum_{\hat{\Delta},s}\,b_{V,\hat{\Delta},s}^{2}\,\hat{f}_{\hat{\Delta},s}(z,\bar{z})\,,\\\
F_{2}(z,\bar{z})=\left(\frac{\sqrt{z\bar{z}}}{(1-z)(1-\bar{z})}\right)^{\Delta_{\phi}}\sum_{\Delta,\ell}\,a_{\mathcal{O}_{T}}\,&\lambda_{\phi\phi\mathcal{O}_{T}}\,f_{\Delta,\ell}(z,\bar{z})=\\\
&=\sum_{\hat{\Delta},s}\,\left(b_{S,\hat{\Delta},s}^{2}-b_{V,\hat{\Delta},s}^{2}\right)\,\hat{f}_{\hat{\Delta},s}(z,\bar{z})\,,\\\
\end{split}$ (A.5)

where the defect conformal blocks are given by

$\hat{f}_{\hat{\Delta},s}(z,\bar{z})=z^{\frac{\hat{\Delta}-s}{2}}\bar{z}^{\frac{\hat{\Delta}+s}{2}}{}_{2}F_{1}\left(-s,\frac{q}{2}-1,2-\frac{q}{2}-s,\frac{z}{\bar{z}}\right){}_{2}F_{1}\left(\hat{\Delta},\frac{p}{2},\hat{\Delta}+1-\frac{p}{2},z\bar{z}\right),$
(A.6)

with $p=1$ and $q=d-1$ for our case. They factorize in radial coordinates

$\hat{f}_{\hat{\Delta},s}(r,w)=\hat{f}_{\hat{\Delta}}(r)\hat{g}_{s}(w),$ (A.7)

with

$\displaystyle\hat{f}_{\hat{\Delta}}(r)$
$\displaystyle=r^{\hat{\Delta}}\,{}_{2}F_{1}\left(\hat{\Delta},\frac{p}{2},\hat{\Delta}+1-\frac{p}{2},r^{2}\right)\,,$
$\displaystyle\hat{g}_{s}(w)=w^{-s}{}_{2}F_{1}\left(-s,\frac{q}{2}-1,2-\frac{q}{2}-s,w^{2}\right).$
(A.8)

The bulk blocks are not known in a closed form, but can be expressed as a sum
of Harish-Chandra functions [85]

$f_{\Delta,\ell}(z,\bar{z})=f^{HS}_{\Delta,\ell}(z,\bar{z})+\frac{\Gamma(\ell+d-2)\Gamma(-\ell-\frac{d-2}{2})}{\Gamma(\ell+\frac{d-2}{2})\Gamma(-\ell)}\frac{\Gamma(\frac{\ell}{2}+\frac{d-p}{2}-\frac{1}{2})\Gamma(-\frac{\ell}{2}+\frac{1}{2})}{\Gamma(\frac{\ell}{2}+\frac{d}{2}-\frac{1}{2})\Gamma(-\frac{\ell}{2}-\frac{p}{2}+\frac{1}{2})}f^{HS}_{\Delta,2-d-\ell}(z,\bar{z}),$
(A.9)

where $f_{\Delta,\ell}^{HS}(z,\bar{z})$ can be expressed as a double infinite
sum

$\begin{split}f_{\Delta,\ell}^{HS}(z,\bar{z})&=\sum_{m}^{\infty}\sum_{n}^{\infty}[(1-z)(1-\bar{z})]^{\frac{\Delta-\ell}{2}+m+n}h_{n}(\Delta,\ell)h_{m}(1-\ell,1-\Delta)\frac{4^{m-n}}{n!m!}\frac{(\frac{\Delta+\ell}{2})_{n-m}}{\left(\frac{\Delta+\ell}{2}-\frac{1}{2}\right)_{n-m}}\\\
&\textstyle\times{}_{4}F_{3}(-n,-m,\frac{1}{2},\frac{\Delta-\ell}{2}-\frac{d}{2}+1;-\frac{\Delta+\ell}{2}+1-n,\frac{\Delta+\ell}{2}-m,\frac{\Delta-\ell}{2}-\frac{d}{2}+\frac{3}{2};1)\\\
&(1-z\bar{z})^{\ell-2m}\textstyle{}_{2}F_{1}(\frac{\Delta+\ell}{2}-m+n,\frac{\Delta+\ell}{2}-m+n,\Delta+\ell-2(m-n),1-z\bar{z}),\end{split}$
(A.10)

where

$h_{n}(\Delta,\ell)=\frac{\left(\frac{\Delta}{2}-\frac{1}{2},\frac{\Delta}{2}-\frac{p}{2},\frac{\Delta+\ell}{2}\right)_{n}}{\left(\Delta-\frac{d}{2}+1,\frac{\Delta+\ell}{2}+\frac{1}{2}\right)_{n}}.$
(A.11)

##  Appendix B Explicit computation of the one-loop correlator

As we discussed in the main text, the discontinuity of the correlator contains
a non trivial term that gives rise to a very complicated integral in the
dispersion formula. Here we will show the computation of that integral and in
particular we will give a derivation of $I(r,w)$ (5.15), namely the non
trivial contribution to the full correlators (5.16) (5.18) . We start by
writing the non trivial part of the discontinuity (5.14) as a sum of
hypergeometric functions, using the definition of the bulk block (A.10), then
we replace the hypergeometric function with its definition as a sum and
finally we exchange the order of the sums. We find

$\begin{split}&\frac{4\pi\sqrt{rw}\left(F\left(\sin^{-1}\left(\sqrt{r}\sqrt{\frac{r-w}{1-rw}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)-F\left(\sin^{-1}\left(\frac{\sqrt{\frac{r-w}{1-rw}}}{\sqrt{r}}\right)|\frac{(rw-1)^{2}}{(r-w)^{2}}\right)\right)}{r-w}=\\\
&=\sum_{n=0}^{\infty}\frac{i\pi
2^{1-2n}r}{1+2n}\left(\frac{(r-w)(rw-1)}{w}\right)^{n}\,{}_{2}F_{1}\left(n+1,n+1;2n+2;1-r^{2}\right)=\\\
&=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{i\pi
2^{1-2n}r}{1+2n}\left(\frac{(r-w)(rw-1)}{w}\right)^{n}\frac{(1-r^{2})^{m}\left((n+1)_{m}(n+1)_{m}\right)}{m!(2n+2)_{m}}=\\\
&=\sum_{m=0}^{\infty}\frac{2i\pi
r\left(1-r^{2}\right)^{m}}{m+1}{}_{3}F_{2}\left(\frac{1}{2},m+1,m+1;\frac{m}{2}+1,\frac{m}{2}+\frac{3}{2};\frac{(r-w)(rw-1)}{4w}\right).\end{split}$
(B.1)

It turns out that this expansion of the discontinuity can be easily integrated
term by term in the dispersion relation and we can find the function $I(r,w)$.
Indeed, plugging the previous expression into the dispersion relation (3.8)
and changing variable from $w^{\prime}$ to
$x=\Big{(}\frac{(1-rw^{\prime})(w^{\prime}-r)}{4w^{\prime}}\Big{)}$, we find

$\begin{split}\textstyle I(r,w)&=\int_{-\infty}^{0}dx\
\sum_{m=0}^{\infty}\frac{4rw(1-r^{2})^{m}\,_{3}F_{2}\left(\frac{1}{2},m+1,m+1;\frac{m}{2}+1,\frac{m}{2}+\frac{3}{2};x\right)}{(m+1)\left(r^{2}(-w)+rw^{2}+r+4wx-w\right)}=\\\
&=\sum_{m=0}^{\infty}\frac{2^{1-m}rw(1-r^{2})^{m}G_{4,4}^{4,2}\left(\frac{4w}{wr^{2}-\left(w^{2}+1\right)r+w}|\begin{array}[]{c}0,0,\frac{m}{2},\frac{m+1}{2}\\\
-\frac{1}{2},0,m,m\\\
\end{array}\right)}{(m!)\left(r^{2}(-w)+rw^{2}+r-w\right)},\end{split}$ (B.2)

where $G$ is the Meijer G-function and we assumed

$\Re\left(\frac{r\left(-rw+w^{2}+1\right)}{w}\right)\leq
1\lor\frac{r\left(-rw+w^{2}+1\right)}{w}\notin\mathbb{R}.$ (B.3)

To rewrite the previous expression in a closed form we can use the integral
representation of the Meijer G-function

$G_{p,q}^{m,n}\left(z\left|\begin{array}[]{c}a_{1},\ldots,a_{n},a_{n+1},\ldots,a_{p}\\\
b_{1},\ldots,b_{m},b_{m+1},\ldots,b_{q}\\\
\end{array}\right.\right)=\frac{1}{2\pi i}\int_{\mathcal{L}}ds\
\frac{\prod_{k=1}^{n}\Gamma\left(-s-a_{k}+1\right)\prod_{k=1}^{m}\Gamma\left(s+b_{k}\right)z^{-s}}{\prod_{k=n+1}^{p}\Gamma\left(s+a_{k}\right)\prod_{k=m+1}^{q}\Gamma\left(-s-b_{k}+1\right)}.$
(B.4)

Plugging the integral representation representation in (B.2), and exchanging
sum and integral, we find

$\begin{split}&\textstyle I(r,w)=\sum\limits_{m=0}^{\infty}\int ds\
\frac{2^{1-m}rw\left(1-r^{2}\right)^{m}}{\Gamma(m+1)(w-r)(rw-1)}\frac{2^{m+1}\Gamma(1-s)^{2}\Gamma(2s-1)\Gamma(m+s)^{2}}{\Gamma(m+2s)}\left(\frac{4w}{(r-w)(rw-1)}\right)^{-s}=\\\
&=-\int_{\mathcal{L}}ds\ \frac{\pi^{2}r4^{1-s}\csc^{2}(\pi
s)}{2s-1}{}_{2}F_{1}\left(s,s;2s;1-r^{2}\right)\left(\frac{w}{(r-w)(rw-1)}\right)^{1-s}=\\\
&=\textstyle\sum\limits_{n=0}^{\infty}\frac{\partial}{\partial
t}\left(\left(\frac{(r-w)(rw-1)}{w}\right)^{n+t}\frac{4^{-n-t}r}{2n+2t+1}\,_{2}F_{1}\left(n+t+1,n+t+1;2(n+t+1);1-r^{2}\right)\right)=\\\
&=\textstyle\sum\limits_{n=0}^{\infty}\frac{\partial}{\partial
t}\left(\left(\frac{(r-w)(rw-1)}{(1+r)^{2}w}\right)^{n+t}\frac{4r}{(1+r)^{2}(2n+2t+1)}{}_{2}F_{1}\left(\frac{1}{2},n+t+1;n+t+\frac{3}{2};\frac{(1-r)^{2}}{(r+1)^{2}}\right)\right)=\\\
&=\left.\textstyle\frac{\partial}{\partial
t}\left(\left(\frac{(r-w)(rw-1)}{(1+r)^{2}w}\right)^{t}\frac{4r}{(1+r)^{2}(2t+1)}F^{112}_{101}\left(\begin{array}[]{c}1+t:\frac{1}{2};\frac{1}{2}+t,1;\\\
\frac{3}{2}+t:-;1+t;\\\ \end{array}\
\left(\frac{1-r}{1+r}\right)^{2},\frac{(r-w)(rw-1)}{(1+r)^{2}w}\right)\right)\right|_{t=0}.\end{split}$
(B.5)

The final result is expressed in terms of a derivative of a Kampé de Fériet
function [88], which is defined by

$F^{ABB^{\prime}}_{CDD^{\prime}}\left(\begin{array}[]{c}(a):(b);(b^{\prime});\\\
(c):(d);(d^{\prime});\\\ \end{array}\
x,y\right)=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{\prod_{j=1}^{A}(a_{j})_{m+n}\prod_{j=1}^{B}(b_{j})_{m}\prod_{j=1}^{B^{\prime}}(b^{\prime}_{j})_{n}}{\prod_{j=1}^{C}(c_{j})_{m+n}\prod_{j=1}^{D}(d_{j})_{m}\prod_{j=1}^{D^{\prime}}(d^{\prime}_{j})_{n}}\frac{x^{m}}{m!}\frac{y^{n}}{n!}.$
(B.6)

This sum converges absolutely if $A+B=C+D+1$, $A+B^{\prime}=C+D^{\prime}+1$,
$|x|<1$ and $|y|<1$, which in our case, together with (B.3), imply

$0<r<1\land\left((0<w<1\land 0<r\leq w)\lor(w=1\land 0<r<1)\lor\left(w>1\land
0<r\leq\frac{1}{w}\right)\right).$ (B.7)

We can extend the result to all other values by exploiting the symmetry of the
correlator with respect to $r\leftrightarrow\frac{1}{r}$ and
$w\leftrightarrow\frac{1}{w}$ [57].

###  B.1 Generating function of the bulk inversion

The first few orders in the $z=1$ expansion of the one-loop generating
function $C^{t}_{T}(z,\beta)$ (5.31) read

$\begin{split}&C^{t}_{T}(z,\beta)={a^{2}_{\phi}}^{(0)}\Bigg{(}(1-z)^{\Delta_{\phi}}\Bigg{(}\frac{\sqrt{\pi}\Gamma\Big{(}\frac{\beta+1}{2}\Big{)}\Gamma\Big{(}1-\frac{\Delta_{\phi}}{2}\Big{)}^{2}\Gamma\Big{(}\frac{\beta}{2}+\Delta_{\phi}-1\Big{)}}{\Gamma\Big{(}\frac{\beta+2}{4}\Big{)}^{2}\Gamma\Big{(}\frac{1}{4}(\beta-2\Delta_{\phi}+4)\Big{)}\Gamma\Big{(}\frac{1}{4}(\beta+2\Delta_{\phi})\Big{)}}+\\\
&+\Bigg{(}\frac{(z-1)^{2}\Big{(}-\beta^{3}-4\beta^{2}\Delta_{\phi}-6\beta^{2}-4\beta\Delta_{\phi}^{2}-20\beta\Delta_{\phi}-8\beta-4\Delta_{\phi}^{2}-16\Delta_{\phi}\Big{)}}{192(\beta+1)}+\\\
&+\frac{(z-1)\Big{(}\beta^{3}+4\beta^{2}\Delta_{\phi}+4\beta^{2}+4\beta\Delta_{\phi}^{2}+12\beta\Delta_{\phi}+4\beta+4\Delta_{\phi}^{2}+8\Delta_{\phi}\Big{)}}{16(\beta+1)(\beta+2\Delta_{\phi}+2)}+\\\
&+\frac{2}{(z-1)(\beta+2\Delta_{\phi}-2)}-\frac{1}{2}\Big{)}(1-z)^{\frac{\beta}{2}+\Delta_{\phi}}\Bigg{)}\times\\\
&\times\Bigg{(}\frac{\Gamma\Big{(}\frac{\beta}{2}\Big{)}^{4}\sin^{2}\Big{(}\frac{\pi\Delta_{\phi}}{2}\Big{)}}{\pi^{2}\Gamma(\beta-1)\Gamma(\beta)}+\frac{\Big{(}\Delta_{\phi}^{2}+2\Delta_{\phi}\Big{)}(z-1)^{2}\Gamma\Big{(}\frac{\beta}{2}\Big{)}^{4}\sin^{2}\Big{(}\frac{\pi\Delta_{\phi}}{2}\Big{)}}{8\pi^{2}\Gamma(\beta-1)\Gamma(\beta)}+\\\
&-\frac{\Big{(}\Delta_{\phi}^{3}+6\Delta_{\phi}^{2}+8\Delta_{\phi}\Big{)}(z-1)^{3}\Gamma\Big{(}\frac{\beta}{2}\Big{)}^{4}\sin^{2}\Big{(}\frac{\pi\Delta_{\phi}}{2}\Big{)}}{48\pi^{2}\Gamma(\beta-1)\Gamma(\beta)}-\frac{\Delta_{\phi}(z-1)\Gamma\Big{(}\frac{\beta}{2}\Big{)}^{4}\sin^{2}\Big{(}\frac{\pi\Delta_{\phi}}{2}\Big{)}}{2\pi^{2}\Gamma(\beta-1)\Gamma(\beta)}\Bigg{)}\Bigg{)}\Bigg{)}+\\\
&-{a^{2}_{\phi}}^{(1)}\frac{\Gamma\left(\frac{\beta}{4}\right)^{2}(1-z)^{\Delta\phi}\Gamma\left(\frac{\beta}{4}+\frac{\Delta\phi}{2}-\frac{1}{2}\right)}{\Gamma\left(\frac{\beta}{2}-\frac{1}{2}\right)\left(2^{\frac{\beta}{2}-\Delta\phi+1}\Gamma\left(\frac{\Delta\phi}{2}\right)^{2}\right)\Gamma\left(\frac{\beta}{4}-\frac{\Delta\phi}{2}+1\right)}+O((z-1)^{4}),\end{split}$
(B.8)

where $\Delta_{\phi}$ needs to be expanded at one loop. The singlet
contribution is simply $C^{t}_{S}(z,\beta)=\frac{1}{N}C^{t}_{T}(z,\beta)$.

##  Appendix C Feynman diagrams

The bulk two-point function of two fundamental fields in the presence of the
defect $\langle\phi_{i}(x)\phi_{j}(y)\rangle_{D}$ can be calculated
perturbatively by evaluating Feynman diagrams at the fixed point. In
particular, to the first order in $\epsilon$ (hence at one loop) there are two
disconnected contributions, which are given by the one-loop correction to the
bulk propagator and the one-loop correction to the product of one-point
functions

$\phi$$\phi$ $\phi$$\phi$ (C.1)

where the blue double line at the bottom represents the defect $D$, whereas
black lines are scalar propagators with a generic one-loop correction
represented as a grey circle. These two contributions naturally provide the
corrections to the defect and bulk identity in (5.16) and (5.18). The only
connected diagram is the cross diagram

$\phi_{i}$$\phi_{j}$$\phi_{1}$$\phi_{1}$ (C.2)

The contribution of the diagram (C.2) is given by

$-\mathcal{N}_{\phi}^{\,2}\left(\delta_{ij}+2\delta_{i1}\delta_{j1}\right)\frac{\lambda
h^{2}}{2!}\int d\tau_{1}\int d\tau_{2}\int d^{4-\epsilon}q\
G(x-q)G(y-q)G(q-x(\tau_{1}))G(q-x(\tau_{2})),$ (C.3)

where $|\dot{x}(\tau_{1,2})|=1$, $\mathcal{N}_{\phi}$ is chosen to normalize
the bulk two-point function in the absence of the defect, and $G(x)$ is the
free propagator

$G(x)=\frac{1}{(d-2)\Omega_{d-1}|x|^{d-2}}=\frac{1}{4\pi^{2}|x|^{2}}+O(\epsilon),\quad\mathcal{N}_{\phi}=2\pi+O(\epsilon).$
(C.4)

Since at the fixed point $\lambda_{*}\propto\epsilon$, we can set $\epsilon=0$
in all the other terms. Evaluating at the fixed point, Inserting (C.4) into
(C.3) and performing the $\tau_{2}$ and $\tau_{2}$ integrals one gets

$-\left(\delta_{ij}+2\delta_{i1}\delta_{j1}\right)\frac{3}{8\pi^{2}}\int
d^{4}q\,\frac{1}{|q_{\perp}|^{2}\,|x-q|^{2}\,|y-q|^{2}}=\left(\delta_{ij}+2\delta_{i1}\delta_{j1}\right)\frac{I(x,y)}{8|x_{\perp}||y_{\perp}|},$
(C.5)

where $I(x,y)$ is exactly the conformally invariant function defined in
(5.15), but expressed in the coordinates $(x,y)$. Thanks to conformal
invariance, it is enough to evaluate $I(x,y)$ for
$x_{\parallel}=y_{\parallel}=0$. However, the integral in (C.5) is very hard
to solve analytically and a general closed form has not been found.
Nevertheless, it is still possible to reduce it to a more useful one-
dimensional integral representation which can be used to obtain series
expansions and other analytic considerations. Moreover, it is also possible to
obtain an exact result for particular regimes, such as
$x_{\perp}^{2}+y_{\perp}^{2}=1$ or $x_{\perp}\parallel y_{\perp}$ (in
lightcone coordinates, $z\bar{z}=1$ and $z=\bar{z}$ respectively).

We begin with the derivation of the integral representation. The starting
point is

$I(x,y)=-\frac{3}{\pi^{2}}\int d^{3}q_{\perp}\int
dq_{\parallel}\,\frac{|x_{\perp}||y_{\perp}|}{|q_{\perp}|^{2}\,(q_{\parallel}^{2}+|x_{\perp}-q_{\perp}|^{2})\,(q_{\parallel}^{2}+|y_{\perp}-q_{\perp}|^{2})}.$
(C.6)

One can introduce a Feynman parameter $\alpha$ for the last two factor in the
denominator

$I(x,y)=-\frac{3}{\pi^{2}}\int_{0}^{1}d\alpha\int d^{3}q_{\perp}\int
dq_{\parallel}\,\frac{|x_{\perp}||y_{\perp}|}{|q_{\perp}|^{2}\,\left(q_{\parallel}^{2}+\alpha|x_{\perp}-q_{\perp}|^{2}+(1-\alpha)|y_{\perp}-q_{\perp}|^{2}\right)^{2}}.$
(C.7)

We can now perform the $q_{\parallel}$ integral and rearrange to obtain

$I(x,y)=-\frac{3}{2\pi}\int_{0}^{1}d\alpha\int
d^{3}q_{\perp}\,\frac{|x_{\perp}||y_{\perp}|}{|q_{\perp}|^{2}\,\left(|q_{\perp}-\tilde{q}_{\perp}|^{2}+L^{2}\right)^{\frac{3}{2}}},$
(C.8)

where

$\tilde{q}_{\perp}=\alpha x_{\perp}+(1-\alpha)y_{\perp},\quad
L^{2}=\alpha(1-\alpha)|x_{\perp}-y_{\perp}|^{2}.$ (C.9)

Introducing another Feynman parameter $\xi$ one gets

$I(x,y)=-\frac{9}{4\pi}\int_{0}^{1}d\alpha\int_{0}^{1}d\xi\int
d^{3}q_{\perp}\,\frac{\xi^{\frac{1}{2}}\cdot|x_{\perp}||y_{\perp}|}{\left(|q_{\perp}-\xi\tilde{q}_{\perp}|^{2}+\xi(1-\xi)|\tilde{q}_{\perp}|^{2}+\xi
L^{2}\right)^{\frac{5}{2}}}.$ (C.10)

After the shift $q_{\perp}\rightarrow q_{\perp}+\xi\tilde{q}_{\perp}$ the
integral over $q_{\perp}$ becomes easy and it gives

$I(x,y)=-3|x_{\perp}||y_{\perp}|\int_{0}^{1}d\alpha\int_{0}^{1}d\xi\,\frac{1}{\xi^{\frac{1}{2}}\left(L^{2}+(1-\xi)|\tilde{q}_{\perp}|^{2}\right)}.$
(C.11)

It is now convenient to pass to lightcone coordinates. Changing variable
$\xi=\eta^{2}$ and rearranging one gets

$I(z,\bar{z})=-6\sqrt{z\bar{z}}\int_{0}^{1}d\alpha\int_{0}^{1}d\eta\,\frac{1}{\left(1+\alpha(z\bar{z}-1)-\eta^{2}(1+(z-1)\alpha)(1+(\bar{z}-1)\alpha)\right)}.$
(C.12)

At this point one can obtain two different representations by integrating
either in $\alpha$ or in $\eta$. Integrating over $\alpha$ one gets

$I(z,\bar{z})=-6\sqrt{z\bar{z}}\int_{0}^{1}d\eta\
\frac{\log\left[P(z,\bar{z},\eta)+\sqrt{Q(z,\bar{z},\eta})\right]-\log\left[P(z,\bar{z},\eta)-\sqrt{Q(z,\bar{z},\eta})\right]}{\sqrt{Q(z,\bar{z},\eta})},$
(C.13)

where $P(z,\bar{z},\eta)$ and $Q(z,\bar{z},\eta)$ are the following
polynomials

$\begin{split}P(z,\bar{z},\eta)&=1+z\bar{z}-\eta^{2}(z+\bar{z}),\\\
Q(z,\bar{z},\eta)&=(z\bar{z}-1)^{2}-2\eta^{2}(z+\bar{z}+z\bar{z}(z+\bar{z}-4))+\eta^{4}(z-\bar{z})^{2}.\end{split}$
(C.14)

Alternatively, it is possible to perform the integral over $\eta$ in (C.12) to
get

$I(z,\bar{z})=-6\sqrt{z\bar{z}}\int_{0}^{1}d\alpha\
\frac{\tanh^{-1}\left[\frac{(1+(z-1)\alpha)(1+(\bar{z}-1)\alpha}{1+\alpha(z\bar{z}-1)}\right]}{\sqrt{(1+(z-1)\alpha)(1+(\bar{z}-1)\alpha)(1+\alpha(z\bar{z}-1))}}.$
(C.15)

Both expressions (C.13) and (C.15) can be used to obtain series expansions of
$I(z,\bar{z})$. Moreover, from (C.15) it is also possible to extract in a
closed form the term in $I(z,\bar{z})$ proportional to $\log(1-\bar{z})$.
Indeed in the limit $\bar{z}\rightarrow 1$ the argument of the hyperbolic
arcotangent goes to $1$ giving a logarithmic divergence, whereas the
denominator remains finite. The logarithmic term is therefore given by

$I(z,\bar{z})_{\log(\bar{z}-1)}=3\sqrt{z\bar{z}}\,\log(\bar{z}-1)\int_{0}^{1}d\alpha\
\frac{1}{\sqrt{(1+(z-1)\alpha)(1+(\bar{z}-1)\alpha)(1+\alpha(z\bar{z}-1))}}.$
(C.16)

The result of this integral can be expressed in terms of incomplete elliptic
integrals of the first kind $F(\varphi,k)$

$\begin{split}I&(z,\bar{z})_{\log(\bar{z}-1)}=\\\
&=6\sqrt{z\bar{z}}\,\log(\bar{z}-1)\left(\frac{F\left(\sin^{-1}\left(\sqrt{\frac{\bar{z}-z}{\bar{z}-1}}\right),\frac{z(\bar{z}-1)^{2}}{(\bar{z}-z)(z\bar{z}-1)}\right)-F\left(\sin^{-1}\left(\sqrt{\frac{\bar{z}-z}{z\bar{z}-z}}\right),\frac{z(\bar{z}-1)^{2}}{(\bar{z}-z)(z\bar{z}-1)}\right)}{\sqrt{(\bar{z}-z)(z\bar{z}-1)}}\right).\end{split}$
(C.17)

###  C.1 Evaluation of the integral for $z=\bar{z}$

We now proceed to solve exactly the integral for some specific regimes. The
first computation is for the case $x_{\perp}\parallel y_{\perp}$ with
$x_{\perp}\cdot y_{\perp}>0$. Again we start from the integral

$I(x,y)=-\frac{3}{\pi^{2}}\int d^{3}q_{\perp}\int
dq_{\parallel}\,\frac{|x_{\perp}||y_{\perp}|}{|q_{\perp}|^{2}\,(q_{\parallel}^{2}+|x_{\perp}-q_{\perp}|^{2})\,(q_{\parallel}^{2}+|y_{\perp}-q_{\perp}|^{2})}.$
(C.18)

Thanks to the conformal invariance of this integral, this is equivalent to
compute it for $x_{\perp}=y_{\perp}$ with $|x_{\perp}|=1$ and
$d=x_{\parallel}-y_{\parallel}$ arbitrary (without loss of generality, we
assume $d>0$). Indeed if $y_{\perp}=\alpha x_{\perp}$ with
$x_{\parallel}=y_{\parallel}=0$, then one can get the same cross ratios by
taking $x_{\perp}=y_{\perp}$ with $|x_{\perp}|=1$ and
$d=\frac{1-\alpha}{\sqrt{\alpha}}$. Hence we can instead solve the integral

$I(x,y)=-\frac{3}{\pi^{2}}\int d^{3}q_{\perp}\int
dq_{\parallel}\,\frac{1}{|q_{\perp}|^{2}\,(q_{\parallel}^{2}+|x_{\perp}-q_{\perp}|^{2})\,((q_{\parallel}^{2}+d)+|x_{\perp}-q_{\perp}|^{2})}.$
(C.19)

Integration over $q_{\parallel}$ is straightforward and gives

$I(x,y)=-\frac{3}{2\pi}\int d^{3}q_{\perp}\
\frac{1}{|q_{\perp}|^{2}}\cdot\frac{1}{|q_{\perp}+x_{\perp}|}\cdot\frac{1}{|q_{\perp}+x_{\perp}|^{2}+\left(\frac{d}{2}\right)^{2}}.$
(C.20)

We can still exploit rotational invariance to set $x_{\perp}=(1,0,0)^{T}$. For
convenience, we also set $a=\frac{d}{2}$. Passing to spherical coordinates
$(\rho,\theta,\phi)$ and performing the longitudinal integral over $\phi$ we
are left with

$I(a)=-3\,\int_{0}^{\infty}d\rho\int_{0}^{\pi}d\theta\frac{\sin\theta}{\sqrt{\rho^{2}+1+2\rho\cos\theta}\
(\rho^{2}+1+2\rho\cos\theta+a^{2})}.$ (C.21)

We change integration variable $\theta\rightarrow t=\cos\theta$ to get

$I(a)=-3\,\int_{0}^{\infty}d\rho\,\frac{1}{(2\rho)^{3/2}}\int_{-1}^{+1}dt\frac{1}{\sqrt{A+t}\
(B+t)},$ (C.22)

with

$A=\frac{\rho^{2}+1}{2\rho},\ \ B=\frac{\rho^{2}+1+a^{2}}{2\rho}.$ (C.23)

The $t$ integral can now be explicitly performed, yielding

$I(a)=-3\,\int_{0}^{\infty}d\rho\
\frac{\arctan{\frac{|\rho+1|}{a}}-\arctan{\frac{|\rho-1|}{a}}}{\rho\cdot a}.$
(C.24)

Using the trigonometric formula for $\arctan u-\arctan v$ and splitting the
domain of integration into $(0,1)$ and $(1,\infty)$ we get

$I(a)=-\frac{3}{a}\,\left[\int_{0}^{1}\frac{d\rho}{\rho}\
\arctan\left(\frac{2\rho
a}{a^{2}+1-\rho^{2}}\right)+\int_{1}^{\infty}\frac{d\rho}{\rho}\
\arctan\left(\frac{2a}{a^{2}+\rho^{2}-1}\right)\right].$ (C.25)

This integration can be performed as well and it gives

$\begin{split}I(a)=&-\frac{3i}{8a}\left[\log\left(-\frac{1}{(i+a)^{2}}\right)\left(\log\left(-\frac{1}{(i+a)^{2}}\right)+2\log\left(a(2i+a)\right)\right)+\right.\\\
&\left.-\log\left(-\frac{1}{(-i+a)^{2}}\right)\left(\log\left(-\frac{1}{(-i+a)^{2}}\right)+2\log\left(a(-2i+a)\right)\right)+\right.\\\
&+4\,\mathrm{Li}_{2}\left(-\frac{i}{-i+a}\right)-4\,\mathrm{Li}_{2}\left(\frac{i}{-i+a}\right)-2\,\mathrm{Li}_{2}\left(1+\frac{1}{(-i+a)^{2}}\right)+\\\
&\left.-4\,\mathrm{Li}_{2}\left(\frac{i}{i+a}\right)+4\,\mathrm{Li}_{2}\left(-\frac{i}{i+a}\right)+2\,\mathrm{Li}_{2}\left(1+\frac{1}{(i+a)^{2}}\right)\right].\end{split}$
(C.26)

Even though in this form the result involves complex numbers, it is of course
real. We can simplify the first two lines using principal logarithm identities
and functional identities for the dilogarithm. After some algebraic
manipulations, massive simplifications occur, and the result is

$I(a)=\frac{6}{a}\cdot\left(\frac{1}{2}\log a^{2}\cdot\arctan
a-\mathrm{Ti}_{2}\,(a)\right),$ (C.27)

where

$\mathrm{Ti}_{2}(\,a)=\frac{\mathrm{Li}_{2}\,(ia)-\mathrm{Li}_{2}\,(-ia)}{2i}=\int_{0}^{a}dt\
\frac{\arctan t}{t}.$ (C.28)

Finally, we can express the integral (C.18) in terms of lightcone coordinates
(recall that in this regime $z=\bar{z}$) just by making the identification
$2a=\frac{1-z}{\sqrt{z}}$

$I(z=\bar{z}>0)=12\,\frac{\sqrt{z}}{1-z}\,\left(\frac{1}{2}\log\left(\frac{(1-z)^{2}}{4z}\right)\cdot\arctan\left(\frac{1-z}{2\sqrt{z}}\right)-\mathrm{Ti}_{2}\,\left(\frac{1-z}{2\sqrt{z}}\right)\right).$
(C.29)

The very same method can be used to obtain a closed form expression for
$x_{\perp}\parallel y_{\perp}$ with $x_{\perp}\cdot y_{\perp}<0$, or
equivalently, $z=\bar{z}<0$. At the end of the calculation one finds

$\begin{split}I(z=\bar{z}<0)&=\frac{3}{2}\,\frac{\sqrt{-z}}{1-z}\left(2\pi^{2}+8\text{Li}_{2}\left(\frac{2\sqrt{-z}-\sqrt{z^{2}-6z+1}}{1-z}\right)-8\text{Li}_{2}\left(-\frac{2\sqrt{-z}-\sqrt{z^{2}-6z+1}}{1-z}\right)-\right.\\\
&\left.-2\text{Li}_{2}\left(\frac{1-z-\sqrt{z^{2}-6z+1}}{\left(1-\sqrt{-z}\right)^{2}}\right)+2\text{Li}_{2}\left(\frac{1-z-\sqrt{z^{2}-6z+1}}{\left(1+\sqrt{-z}\right)^{2}}\right)+\right.\\\
&\left.+2\text{Li}_{2}\left(\frac{\left(1-\sqrt{-z}\right)^{2}}{1-z+\sqrt{z^{2}-6z+1}}\right)-2\text{Li}_{2}\left(\frac{\left(1+\sqrt{-z}\right)^{2}}{1-z+\sqrt{z^{2}-6z+1}}\right)+\right.\\\
&\left.+4\log\left(\frac{1-z+\sqrt{z^{2}-6z+1}}{2\sqrt{-z}}\right)\left(2\log\left(\frac{1-z}{2\sqrt{-z}}\right)-2\log\left(1+\frac{\sqrt{z^{2}-6z+1}}{2\sqrt{-z}}\right)-\right.\right.\\\
&\left.\left.-\frac{1}{2}\log\left(\left(\frac{1-\sqrt{-z}}{1+\sqrt{-z}}\right)^{2}\right)\right)-\log\left(\frac{(z+1)^{2}}{-4z}\right)\log\left(\left(\frac{1-\sqrt{-z}}{1+\sqrt{-z}}\right)^{2}\right)\right).\end{split}$
(C.30)

###  C.2 Evaluation of the integral for $z\bar{z}=1$

At last, we also derive a closed form for the integral in the case
$x_{\perp}^{2}+y_{\perp}^{2}=1$ ($z\bar{z}=1$). We start again from

$I(x,y)=-\frac{3}{\pi^{2}}\int d^{3}q_{\perp}\int
dq_{\parallel}\,\frac{|x_{\perp}||y_{\perp}|}{|q_{\perp}|^{2}\,(q_{\parallel}^{2}+|x_{\perp}-q_{\perp}|^{2})\,(q_{\parallel}^{2}+|y_{\perp}-q_{\perp}|^{2})}.$
(C.31)

Integration over $q_{\parallel}$ is straightforward and yields

$I(x,y)=-\frac{3}{\pi}|x_{\perp}||y_{\perp}|\int
d^{3}q_{\perp}\,\frac{1}{|q_{\perp}|^{2}}\cdot\frac{1}{|x_{\perp}-q_{\perp}|\cdot|y_{\perp}-q_{\perp}|\cdot\left(|x_{\perp}-q_{\perp}|+|y_{\perp}-q_{\perp}|\right)}.$
(C.32)

Now we use the partial fraction decomposition

$\frac{1}{A\cdot
B\cdot(A+B)}=\frac{1}{B\cdot(A^{2}-B^{2})}-\frac{1}{A\cdot(A^{2}-B^{2})},$
(C.33)

to get two integrals

$\begin{split}I(x,y)=-\frac{3}{\pi}|x_{\perp}||y_{\perp}|\bigg{(}\int
d^{3}q_{\perp}&\,\frac{1}{|q_{\perp}|^{2}}\cdot\frac{1}{|y_{\perp}-q_{\perp}|\left(|x_{\perp}-q_{\perp}|^{2}-|y_{\perp}-q_{\perp}|^{2}\right)}-\\\
&-\int
d^{3}q_{\perp}\,\frac{1}{|q_{\perp}|^{2}}\cdot\frac{1}{|x_{\perp}-q_{\perp}|\left(|x_{\perp}-q_{\perp}|^{2}-|y_{\perp}-q_{\perp}|^{2}\right)}\bigg{)},\end{split}$
(C.34)

The second term can be reduced to the one of the first in the following
manner: let $M\in GL(3)$ such that $x_{\perp}=My_{\perp}$. $M$ can be
decomposed as a dilation by $\frac{|x_{\perp}|}{|y_{\perp}|}$ followed by a
rotation in the $x_{\perp}y_{\perp}$ plane, hence
$||M||=\frac{|x_{\perp}|}{|y_{\perp}|}$. Let
$\tilde{x}_{\perp}=M^{-1}y_{\perp}$, so that $y_{\perp}=M\tilde{x}_{\perp}$.
Now we change integration variable: $q_{\perp}=Mq_{\perp}^{\prime}$,
$d^{3}q_{\perp}=||M||^{3}\,d^{3}q_{\perp}^{\prime}=\frac{|x_{\perp}|^{3}}{|y_{\perp}|^{3}}\,d^{3}q_{\perp}^{\prime}$.
Since $M$ is invertible and has only the eigenvalue
$\frac{|x_{\perp}|}{|y_{\perp}|}$, it holds
$|Aq_{\perp}^{\prime}+Az_{\perp}|^{k}=||A||^{k}\cdot|q_{\perp}^{\prime}+z_{\perp}|^{k}$.
After some algebraic manipulations, the second term in (C.34) reduces to

$-\frac{3}{\pi}|x_{\perp}||y_{\perp}|\int
d^{3}q_{\perp}\,\frac{1}{|q_{\perp}|^{2}}\cdot\frac{1}{|y_{\perp}-q_{\perp}|\left(|\tilde{x}_{\perp}-q_{\perp}|^{2}-|y_{\perp}-q_{\perp}|^{2}\right)},$
(C.35)

which is in the same form of the first integral, with $x_{\perp}$ replaced by
$\tilde{x}_{\perp}=M^{-1}y_{\perp}$. Therefore we only need to solve the
integral of the first term. Moreover, using conformal invariance, we can set
$y_{\perp}=(1,0,0)^{T}$ and $x_{\perp}=(x_{1},x_{2},0)^{T}$. Passing to
spherical coordinates $(\rho,\theta,\phi)$ this can be rewritten as

$I(x,y)=-\frac{3}{\pi}|x_{\perp}||y_{\perp}|\int_{0}^{\infty}d\rho\
\int_{0}^{\pi}\sin\theta d\theta\,\int_{0}^{2\pi}d\phi\
\frac{1}{|y_{\perp}-q_{\perp}|}\cdot\frac{1}{|x_{\perp}-q_{\perp}|^{2}-|y_{\perp}-q_{\perp}|^{2}}.$
(C.36)

Now the first fraction can be carried out the integration over $\phi$ since
$|y_{\perp}-q_{\perp}|=\sqrt{\rho^{2}+1+2\rho\cos\theta}$ is independent of
$\phi$, whereas the denominator of the second fraction is of the form
$C+\sin\phi$. Therefore, we can use

$\int_{0}^{2\pi}d\phi\
\frac{1}{C+\sin\phi}=2\pi\frac{\Theta(C^{2}-1)}{\sqrt{C^{2}-1}}\cdot\text{sgn}(C),$

valid for real $C$, where $\Theta(\cdot)$ is the Heaviside step function.
After straightforward algebraic manipulations, changing variable
$\theta\rightarrow t=\cos\theta$, and adding the second piece (C.35), we find
that the integral (C.31) can be rewritten as

$I=J(x_{1},x_{2})+\frac{1}{x_{1}^{2}+x_{2}^{2}}\cdot
J\left(\frac{x_{1}}{x_{1}^{2}+x_{2}^{2}},\frac{-x_{2}}{x_{1}^{2}+x_{2}^{2}}\right),$
(C.37)

with

$J(x_{1},x_{2})=-6\int_{0}^{\infty}d\rho\,\int_{-1}^{+1}dt\
\frac{\text{sgn}(C)}{\sqrt{\rho^{2}+1+2\rho
t}}\cdot\frac{\Theta\left((x_{1}^{2}+x_{2}^{2}-1+2\rho(x_{1}-1)t)^{2}-4x_{2}^{2}\rho^{2}(1-t^{2})\right)}{\sqrt{(x_{1}^{2}+x_{2}^{2}-1+2\rho(x_{1}-1)t)^{2}-4x_{2}^{2}\rho^{2}(1-t^{2})}}.$
(C.38)

This integral is very difficult to handle because of the Heaviside step
function, which is equivalent to integrating over a very complicated domain.
If we restrict to the unit circle $|x_{\perp}|=x_{1}^{2}+x_{2}^{2}=1$ (recall
that we already set $y_{\perp}$ to be a unit vector) the integral can be
considerably simplified

$J(x_{1},x_{2})=-3\int_{0}^{\infty}d\rho\,\int_{-1}^{+1}dt\
\frac{-\text{sgn}(t)}{\rho\cdot\sqrt{\rho^{2}+1+2\rho
t}}\cdot\frac{\Theta\left(2t^{2}(1-x_{1})-x_{2}^{2}\right)}{\sqrt{2t^{2}(1-x_{1})-x_{2}^{2}}}.$
(C.39)

Now we can rewrite the integral over $t$ as the sum of the integrals from $-1$
to $0$ and from $0$ to $1$ so that the $\text{sgn}(t)$ function disappears,
and we can then exchange the order of integration. We also rewrite the
Heaviside step function as a boundary in the integral over $t$.

$J(x_{1},x_{2})=-\frac{3}{|x_{2}|}\,\int_{1/\alpha}^{1}dt\,\frac{1}{\sqrt{\alpha^{2}t^{2}-1}}\int_{0}^{\infty}d\rho\,\frac{1}{\rho}\cdot\left(\frac{1}{\sqrt{\rho^{2}+1-2\rho
t}}-\frac{1}{\sqrt{\rho^{2}+1+2\rho t}}\right),$ (C.40)

where $\alpha=\frac{\sqrt{2(1-x_{1})}}{|x_{2}|}$. Integrating over $\rho$
yields

$J(x_{1},x_{2})=-\frac{3}{|x_{2}|}\,\int_{1/\alpha}^{1}dt\,\frac{1}{\sqrt{\alpha^{2}t^{2}-1}}\log\left(\frac{1+t}{1-t}\right).$
(C.41)

Using $x_{1}^{2}+x_{2}^{2}=1$ and the fact that $J(x_{1},x_{2})$ does not
depend on the sign of $x_{2}$ we find that the second term in (C.37) is
exactly equal to the first, and thus

$I=-\frac{6}{|x_{2}|}\,\int_{1/\alpha}^{1}dt\,\frac{1}{\sqrt{\alpha^{2}t^{2}-1}}\log\left(\frac{1+t}{1-t}\right).$
(C.42)

We can rewrite this in a more convenient way using another change of variable
$t=\frac{a^{2}+s^{2}}{2a^{2}s}$

$I=-\frac{6}{|x_{2}|}\int_{\alpha^{2}-\sqrt{\alpha^{4}-\alpha^{2}}}^{\alpha}ds\,\frac{\log(s^{2}+2\alpha^{2}s+s^{2})-\log(-s^{2}+2\alpha^{2}s-\alpha^{2})}{\alpha
s},$ (C.43)

Finally, solving this integral and passing to lightcone coordinates
$z=e^{i\phi}$, $\bar{z}=e^{-i\phi}$ with $0<\phi<\pi$ (note that it is
possible to extend the result to the whole circle by simply imposing the
symmetry $\phi\rightarrow-\phi$) yields

$\begin{split}I\left(z=e^{i\phi},\bar{z}=e^{-i\phi}\right)&=\frac{3}{4}\csc\left(\frac{\phi}{2}\right)\left(4\text{Li}_{2}\left(\frac{2}{\sin\left(\frac{\phi}{2}\right)+1}-1\right)-4\text{Li}_{2}\left(\frac{\cos(\phi)+4\sin\left(\frac{\phi}{2}\right)-3}{\cos(\phi)+1}\right)\right.+\\\
&+4\text{Li}_{2}\left(-\sec\left(\frac{\phi}{2}\right)-\tan\left(\frac{\phi}{2}\right)\right)-4\text{Li}_{2}\left(\sec\left(\frac{\phi}{2}\right)-\tan\left(\frac{\phi}{2}\right)\right)+\\\
&+4\text{Li}_{2}\left(\tan\left(\frac{\phi}{2}\right)-\sec\left(\frac{\phi}{2}\right)\right)-4\text{Li}_{2}\left(\sec\left(\frac{\phi}{2}\right)+\tan\left(\frac{\phi}{2}\right)\right)+\\\
&+\pi\left(\pi-8i\log\left(\sin\left(\frac{\phi}{4}\right)+\cos\left(\frac{\phi}{4}\right)\right)\right)+8\log\left(\sin\left(\frac{\phi}{4}\right)+\cos\left(\frac{\phi}{4}\right)\right)\times\\\
&\times\left(\log\left(\tan\left(\frac{\phi}{2}\right)\right)-2\tanh^{-1}\left(\frac{4\sin\left(\frac{\phi}{2}\right)+\cos(\phi)-3}{\cos(\phi)+1}\right)\right)+\\\
&+8\log\left(\sec\left(\frac{\phi}{2}\right)\right)\left(\log\left(\frac{2\sin\left(\frac{\phi}{4}\right)}{\sin\left(\frac{\phi}{2}\right)+\cos\left(\frac{\phi}{2}\right)+1}\right)+\right.\\\
&\left.+\left.\tanh^{-1}\left(\tan\left(\frac{\phi}{2}\right)+\sec\left(\frac{\phi}{2}\right)\right)+\coth^{-1}\left(\tan\left(\frac{\phi}{2}\right)+\sec\left(\frac{\phi}{2}\right)\right)\right)\right).\end{split}$
(C.44)

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