Bootstrapping line defects with $O(2)$ global symmetry

Aleix Gimenez-Grau1, Edoardo Lauria2, Pedro Liendo1 and Philine van Vliet1

1 DESY Hamburg, Theory Group,

Notkestraße 85, D-22607 Hamburg, Germany
2 Centre de Physique Théorique (CPHT), Ecole Polytechnique

91128 Palaiseau Cedex, France

<EMAIL_ADDRESS><EMAIL_ADDRESS>
aleix.gimenez<EMAIL_ADDRESS>

We use the numerical bootstrap to study conformal line defects with $O(2)$
global symmetry. Our results are very general and capture in particular
conformal line defects originating from bulk CFTs with a continuous global
symmetry, which can either be preserved or partially broken by the presence of
the defect. We begin with an agnostic approach and perform a systematic
bootstrap study of correlation functions between two canonical operators on
the defect: the displacement and the tilt. We then focus on two interesting
theories: a monodromy line defect and a localized magnetic field line defect.
To this end, we combine the numerical bootstrap with the
$\varepsilon$-expansion, where we complement existing results in the
literature with additional calculations. For the monodromy defect our
numerical results are consistent with expectations, with known analytic
solutions sitting inside our numerical bounds. For the localized magnetic
field line defect our plots show a series of intriguing cusps which we
explore.

###### Contents

  1. 1 Introduction
  2. 2 Line defects with global symmetry
    1. 2.1 Discrete symmetries and parity
    2. 2.2 Universal defect operators
    3. 2.3 Crossing equations
  3. 3 Defect theories in the $\varepsilon$-expansion
    1. 3.1 Monodromy defects with $SO(2)_{F}$ symmetry
    2. 3.2 Localized magnetic field line defect
  4. 4 Numerical results
    1. 4.1 Bounds on universal correlators
    2. 4.2 Bootstrapping the monodromy line defect in the $O(2)$ model
    3. 4.3 Bootstrapping the localized magnetic field line defect
  5. 5 Conclusions and future directions
  6. A More data on the $O(N)$ monodromy defect in 3d
    1. A.1 OPE coefficients with the displacement operator
    2. A.2 The leading singlet
  7. B Crossing vectors
    1. B.1 One complex scalar
    2. B.2 Tilt and displacement
    3. B.3 One real and one complex scalar
    4. B.4 Two complex scalars

##  1 Introduction

Defects offer an ideal bridge between low-energy and high-energy physics,
experiments and theory. On the one hand defects such as boundaries and
impurities are common in low-energy systems, like ferromagnetic materials and
spin systems, where they induce a variety of curious phenomena, including the
famous Kondo effect [1, 2, 3, 4, 5, 6]. On the other hand, line and surface
operators, boundary conditions and interfaces offer helpful theoretical tools
to investigate the structure of Quantum Field Theories (QFTs). Prototypical
examples are Wilson and ’t Hooft operators in 4d gauge theories, which provide
the order parameter for confinement [7, 8, 9], and vortices in superfluids and
superconductors, which can confine or proliferate, signaling distinct phases
of the bulk [10]. Beside specific examples, the language of defects can be
used to characterize generalized global symmetries [11], to investigate
dualities between 4d gauge theories [12], and to study information-theoretic
aspects of QFTs [13, 14]. Furthermore, defects offer alternative strategies to
engineer lower-dimensional strongly-coupled QFTs, for example Gaiotto-Witten
$\mathcal{T}[G]$ theories [15, 16] as conformal boundary conditions for
$\mathcal{N}=4$ SYM, 3d abelian gauge theories as conformal boundary
conditions for the 4d Maxwell field [17], and $O(N)$ vector models as
conformal boundary conditions for the 4d free massless scalar field [18].
Finally, defects with higher co-dimensions can be used to obtain higher-
dimensional descriptions of theories with long-range interactions, for example
the long-range Ising model [19], as well as the long-range $O(N)$ models [20].

In this paper we explore the space of co-dimension two conformal line defects
with $O(N)$ global symmetry. We base our approach on the modern conformal
bootstrap of [21] (see e.g. [22, 23, 24] for three recent reviews on the
subject), and we focus exclusively on the 1d theory living on the line defect.
This work is then a natural extension of [25], where line defects with a
$\mathbb{Z}_{2}$ global symmetry were studied using similar techniques. Other
setups where this strategy has been successful include line defects in
supersymmetric models in [26, 27] and in the study of the long-range Ising
model in various dimensions in [28]. More recently it has also been employed
in order to carve out the space of conformal boundary conditions for a free
massless scalar in [29, 30].

By definition, a co-dimension two conformal line defect with $O(N)$ global
symmetry preserves a ‘little’ conformal group $SL(2,\mathbb{R})$ along the
line times a residual ‘transverse rotations’ group $SO(2)_{T}$ about the
line.111Such ‘transverse rotations’ may be broken e.g. by spinning conformal
defects, see e.g. [31]. We can think of $O(N)$ to be the remnant bulk global
symmetry after the introduction of the defect in the homogeneous 3d CFT with a
global symmetry $G$ so that

$\displaystyle SO(4,1)\times G\longrightarrow SL(2,\mathbb{R})\times
SO(2)_{T}\times O(N)~{},\quad O(N)\subseteq G~{}.$ (1.1)

The residual symmetry of the theory allows us to define local defect operators
that are primaries with respect to the ‘little’ conformal group. Such
operators will then be conveniently labeled by their scaling dimensions
$\Delta$, which are non-negative in unitary theories, their transverse
$SO(2)_{T}$ spins $s$ and their $SO(N)$ charges $r_{i}$. Since all connected
rotations along the line are trivial, we just need to distinguish between
parity-odd and parity-even scalar defect operators, whenever parity on the
line is preserved.222In the context of the 3d Ising model, this parity has
been called $\mathcal{S}$-parity [25, 2].

Correlation functions between defect operators are akin to those of a one-
dimensional CFT and must be crossing symmetric in the usual sense. Hence,
while on the one hand these correlation functions enjoy ‘positivity’ in
unitary theories and so they can be bootstrapped using semi-definite
programming,333See [32, 33] for an alternative numerical bootstrap method that
does not require positivity. on the other hand they know little about the
bulk. While it remains a very interesting open problem to understand how, for
generic bulk CFTs, the bulk information can be encoded into defect correlation
functions,444Some progress in this direction can be found in [34, 29, 30] in
the context of conformal boundaries and defects for the free massless scalar
field and in [35] in the context of surface defects for the 4d Maxwell field.
at the same time we find that a systematic exploration of co-dimension two
line defects with $O(N)$ global symmetry purely based on the numerical
conformal bootstrap technique is still missing.555See [36] and references
therein for works that combine analytic functionals in 1d with the numerical
conformal bootstrap. In the present paper we aim at filling this gap, and in
doing so we will be starting from co-dimension two defects with $O(2)_{F}$
global symmetry.666Here and below (whenever necessary) we will denote the
global symmetry group as $O(2)_{F}$, to be distinguished from the group of
transverse rotations about the defect that is denoted as $SO(2)_{T}$.

In section 4 we will present a systematic bootstrap study of certain universal
defect observables such as mixed correlation functions involving the
displacement operator and the tilt operator. As we will review in section 2,
these observables capture the breaking of certain bulk local symmetries by the
presence of the defect. Along with the agnostic bootstrap and whenever
possible, we will try to isolate known defect theories by making gap
assumptions inspired by $\varepsilon$-expansion predictions in specific
models. There are indeed quite a few instances of interesting conformal
defects of this sort which should be allowed by our bootstrap bounds. The
preeminent example is the so-called localized magnetic field line defect or
magnetic line defect (see [37, 5] and references therein), which for our
purposes can be defined in the $O(3)$ CFT and breaks the bulk global symmetry
down to $O(2)_{F}$, i.e.

$\displaystyle
S_{\text{LML}}=S_{O(3)}+h\int_{-\infty}^{\infty}d\tau\,\phi_{1}(\tau)~{},$
(1.2)

where $\phi_{1}$ is one component of the fundamental $O(3)$ vector. This
symmetry breaking implies the existence of a tilt in the spectrum, namely a
defect primary operator with protected scaling dimension $\Delta_{t}=1$ and
transforming as a vector of $O(2)_{F}$. A review of known results on the
magnetic line defect, along with original computations of correlators relevant
to our study in the $\varepsilon$-expansion will be presented in section 3.
Other interesting known examples of co-dimension two line defects with
continuous global symmetry are of the monodromy type, i.e. they can be thought
of as boundaries of co-dimension one topological operators that implement a
bulk global symmetry transformation $g\in G$, see e.g. [2, 25]. For our
purposes we can consider a bulk CFT with global symmetry $G$ and a complex
scalar field $\Phi$ charged under $U(1)_{F}\in G$. We can then define an
$U(1)_{F}$-preserving monodromy defect for any element $g=e^{2\pi iv}\in
U(1)_{F}$ by requiring $\Phi$ to be single valued after a $SO(2)_{T}$ rotation
only up to $g$, i.e. [38, 39, 40]

$\Phi(r,\theta+2\pi,\vec{x})=e^{2\pi iv}\Phi(r,\theta,\vec{x})\>,\quad v\sim
v+1~{},\quad v\in[0,1)~{},$ (1.3)

where $(r,\theta)$ are the polar coordinates in the transverse plane with
respect to the defect. For both $v=0$ (i.e. no monodromy) and $v=1/2$, the
internal $U(1)_{F}$ symmetry of $\Phi$ will get enhanced to $O(2)_{F}$, which
includes complex conjugation of $\Phi$. Now, the specific choice for the
monodromy has broken the original symmetry $G$ down to $U(1)_{F}$, and
therefore the resulting defect may feature a tilt operator as well. In our
study however we will focus on the $O(2)$ bulk global symmetry, for which the
defect spectrum does not contain a tilt operator. We will present a small
review on $\varepsilon$-expansion results relevant to our study in section 3.

##  2 Line defects with global symmetry

In this section we discuss some universal properties of co-dimension two line
defects with global symmetry. We start our analysis with a discussion of the
discrete symmetries that characterize a line defect. We then introduce the two
main characters of our bootstrap analysis: the displacement operator and the
tilt operator. We conclude by presenting the crossing equations for the
correlators that we will study in section 4.

###  2.1 Discrete symmetries and parity

In addition to the continuous part of the symmetry, e.g. the group in eq.
(1.1), we are also interested in the case where the symmetry group involves
improper reflections along the parallel or transverse directions with respect
to the line, i.e.

$\displaystyle O^{+}(2,1)\times O(2)_{T}\times O(2)_{F}\>,\qquad
O^{+}(2,1)\times SO(2)_{T}\times O(2)_{F}\>.$ (2.1)

We can think about the $O^{+}(2,1)$ parity as the improper rotations in 1d,
such that spin-odd primaries in higher dimensions become parity-odd in 1d.
This is the $\mathcal{S}$-parity of [2, 25],

$\displaystyle\mathcal{S}:\qquad\tau\rightarrow-\tau~{},\quad\mathcal{S}(\psi(\tau))=(-1)^{S_{\psi}}\psi(-\tau)~{},\quad
S_{\psi}=0,1~{},$ (2.2)

so that invariance under $\mathcal{S}$-parity of defect correlation functions
implies that $(\tau_{i}<\tau_{i+1})$

$\displaystyle\langle\psi_{1}(\tau_{1})\psi_{2}(\tau_{2})\psi_{3}(\tau_{3})\rangle$
$\displaystyle=(-1)^{S_{1}+S_{2}+S_{3}}\langle\psi_{3}(-\tau_{3})\psi_{2}(-\tau_{2})\psi_{1}(-\tau_{1})\rangle\>.$
(2.3)

For the conformal three-point correlation functions, invariance under
$\mathcal{S}$-parity means that [25]

$\lambda_{123}=(-1)^{S_{1}+S_{2}+S_{3}}\lambda_{213}~{}.$ (2.4)

Hence, only $\mathcal{S}$-parity even operators are allowed to appear in the
fusion of two identical local defect operators.777Note that the connected
component of the conformal group does not change the cyclic order of operators
insertions, so $\lambda_{123}=\lambda_{231}=\lambda_{312}$. Throughout this
paper we will assume $\mathcal{S}$-parity invariant defects and so
$\mathcal{S}$ will play an important role in our numerical bootstrap study of
section 4.

The second parity assignment is for $O(2)_{T}$. We will adopt the convention
of [25] and denote the action of the $O(2)_{T}$-parity with $\mathcal{B}$. The
action $\mathcal{B}$ is a reflection in a plane perpendicular to the defect
[2] which flips the sign of one of the transverse coordinates888Flipping the
sign of both transverse coordinates would be a transformation with
$\text{det}(\mathcal{B}^{{}^{\prime}})=1$ and is part of the connected part of
the group $O(2)_{T}$ instead of the disconnected part. and reverses the
$O(2)_{T}$ charge (as it follows from the anti-symmetric properties of the
generator of the rotations around the defect, $\mathcal{M}_{xy}$)

$\displaystyle\mathcal{B}:\qquad(x,y)\rightarrow(-x,y)~{},\quad\mathcal{B}(\psi_{s}(\tau))=b_{\psi_{s}}\psi_{-s}(\tau)~{}.$
(2.5)

The coefficient $b_{\psi_{s}}$ determines the parity of the operator. Without
loss of generality we can choose a basis of operators such that operators with
$s\neq 0$ are even under $O(2)_{T}$-parity, while operators with $s=0$ can be
both even or odd [25]. Throughout this paper we will not require $O(2)_{T}$
parity to be a symmetry of the defect CFT.

###  2.2 Universal defect operators

####  2.2.1 The displacement operator

When considering a local $d$-dimensional bulk CFT, i.e. with a stress-energy
tensor $T^{\mu\nu}$, conservation of $T^{\mu\nu}$ is generically violated by
terms localized on the defect [25, 41]:

$\displaystyle\partial_{\mu}T^{\mu
i}=-\delta^{(q)}(\mathcal{D})\text{D}^{i}~{}.$ (2.6)

Here $\delta^{(q)}(\mathcal{D})$ is a Dirac delta function with support on the
co-dimension $q=d-p$ defect, and $i=1,\ldots,q$ is an index in the directions
orthogonal to the defect. The operator D on the right-hand side of the eq.
(2.6) is the displacement operator, i.e. a defect primary operator of scaling
dimension $\Delta=p+1$ and a vector under $SO(2)_{T}$. For the co-dimension
two case we use the notation

$\displaystyle\text{D}\equiv\text{D}^{1}+i\text{D}^{2}~{},\quad\bar{\text{D}}\equiv\text{D}^{1}-i\text{D}^{2}\>,$
(2.7)

to denote the positive and negative spin components.

When taking the correlator of four displacement operators, there are two OPEs
that we need to consider. In a theory where $\mathcal{S}$-parity is preserved,
the $\text{D}\times\bar{\text{D}}$ OPE contains the identity operator
${\mathds{1}}$, as well as $SO(2)_{T}$ singlet operators, which can be either
even or odd under $\mathcal{S}$-parity as dictated by eq. (2.4). We denote
these operators as ${(\text{D}\bar{\text{D}})}^{\pm}$. The
$\text{D}\times\text{D}$ OPE exchanges operators with transverse spin $s=2$
and positive $\mathcal{S}$-parity, denoted ${\text{D}^{2}}$. All in all, we
get

$\text{D}\times\bar{\text{D}}\sim{\mathds{1}}+{(\text{D}\bar{\text{D}})}^{+}+{(\text{D}\bar{\text{D}})}^{-}+\cdots\>,\quad\text{D}\times\text{D}\sim{\text{D}^{2}}+\cdots\>.$
(2.8)

####  2.2.2 The tilt operator

In analogy to the case of the displacement operator, a conformal defect that
breaks a local continuous global symmetry of the bulk must feature a tilt
operator.999Here the term ‘local’ refers to the existence of a conserved
current in the bulk that can be used to define conserved charges. Consider the
case where the bulk global symmetry $G$ is broken down to a subgroup $H$. If
$J^{\mu}_{A}$ is the conserved current of the symmetry $G$, then for each
symmetry generator broken by the defect we have [42, 43, 44]

$\displaystyle\partial_{\mu}J^{\mu}_{A}=$
$\displaystyle\delta^{(q)}(\mathcal{D})t_{A}~{},\quad A\in G/H~{}.$ (2.9)

The tilt $t_{A}$ has protected scaling dimension $\Delta_{t}=p$ and is a
scalar under rotations. Furthermore, the tilt consists of
$\text{dim}(G)-\text{dim}(H)$ components, which are organized into irreducible
representations of the preserved subgroup $H$. The example we consider in this
work is $G=O(N)$ and $H=O(N-1)$, when the tilt transforms in the vector
representation of $H$. For the particular case $N=3$, such that the preserved
sugroup is $O(2)_{F}$, the tilt consists of two components $t^{1}$ and
$t^{2}$, which can be expressed in complex notation as

$t\equiv t^{1}+it^{2}\>,\qquad\bar{t}\equiv t^{1}-it^{2}\>.$ (2.10)

The symmetries allow us to define the following OPEs:

$\displaystyle
t\times\bar{t}\sim{\mathds{1}}+(t\bar{t})^{+}+(t\bar{t})^{-}+\dots,\quad
t\times t\sim t^{2}+\dots\>.$ (2.11)

In the expressions above, the operators $(t\bar{t})^{\pm}$ are $O(2)_{F}\times
SO(2)_{T}$ singlets and even (odd) under $\mathcal{S}$-parity. The operators
$t^{2}$ are charged under $O(2)_{F}$, and they are even under
$\mathcal{S}$-parity.

###  2.3 Crossing equations

We are interested in the study of one-dimensional conformal defects in three-
dimensional bulk CFTs, but because we restrict our attention to four-point
functions on the defect, the crossing equations are identical to the ones for
regular one-dimensional CFTs. In particular, a general four-point function
takes the form

$\displaystyle\langle\phi_{i}(\tau_{1})\phi_{j}(\tau_{2})\phi_{k}(\tau_{3})\phi_{l}(\tau_{4})\rangle=\frac{G_{ijkl}(\xi)}{|\tau_{12}|^{\Delta_{i}+\Delta_{j}}|\tau_{34}|^{\Delta_{k}+\Delta_{l}}}\left(\frac{|\tau_{24}|}{|\tau_{14}|}\right)^{\Delta_{i}-\Delta_{j}}\left(\frac{|\tau_{14}|}{|\tau_{13}|}\right)^{\Delta_{k}-\Delta_{l}}\>,$
(2.12)

where $\tau$ is the coordinate along the defect, and we introduced the cross-
ratio

$\displaystyle\xi=\frac{\tau_{12}\tau_{34}}{\tau_{13}\tau_{24}}\>.$ (2.13)

The external operators are ordered along the line
$\tau_{1}<\tau_{2}<\tau_{3}<\tau_{4}$, such that the cross-ratio takes the
values $0<\xi<1$. In the limit when $\tau_{1}$ approaches $\tau_{2}$, or
equivalently when $\xi\to 0$, the correlation function $G_{ijkl}(\xi)$ admits
an expansion in s-channel conformal blocks [45]

$\displaystyle
G_{ijkl}(\xi)=\sum_{{\mathcal{O}}}\lambda_{ij{\mathcal{O}}}\lambda_{kl{\mathcal{O}}}g_{\Delta}^{\Delta_{ij},\Delta_{kl}}(\xi)\>,\quad
g_{\Delta}^{\Delta_{ij},\Delta_{kl}}(\xi)=\xi^{\Delta}{}_{2}F_{1}(\Delta-\Delta_{ij},\Delta+\Delta_{kl};2\Delta;\xi)\>,$
(2.14)

where $\lambda_{ij{\mathcal{O}}}$ are three-point OPE coefficients. It is
natural to think of the one-dimensional CFT as living on a circle, which is
the conformal compactification of the real line. On the circle, it is clear
that the correlator should respect cyclicity
$\langle\phi_{i}\phi_{j}\phi_{k}\phi_{l}\rangle=\langle\phi_{l}\phi_{i}\phi_{j}\phi_{k}\rangle$,
which, including the prefactor in eq. (2.12), leads to

$\displaystyle(1-\xi)^{\Delta_{j}+\Delta_{k}}G_{ijkl}(\xi)=\xi^{\Delta_{i}+\Delta_{j}}G_{lijk}(1-\xi)\>.$
(2.15)

The crossing equation is obtained by requiring consistency between cyclicity
and the conformal block decomposition. Using standard manipulations, see for
example [46], we can write the crossing equations as

$\displaystyle\sum_{{\mathcal{O}}}\left[\lambda_{ij{\mathcal{O}}}\lambda_{kl{\mathcal{O}}}F_{\mp,\Delta}^{ij,kl}(\xi)\pm\lambda_{kj{\mathcal{O}}}\lambda_{il{\mathcal{O}}}F_{\mp,\Delta}^{kj,il}(\xi)\right]=0\>,$
(2.16)

where $F_{\pm,\Delta}^{ij,kl}$ are defined similarly as for higher-dimensional
CFTs:

$\displaystyle
F_{\pm,\Delta}^{ij,kl}(\xi)\equiv(1-\xi)^{\Delta_{k}+\Delta_{j}}g_{\Delta}^{\Delta_{ij},\Delta_{kl}}(\xi)\pm\xi^{\Delta_{k}+\Delta_{j}}g_{\Delta}^{\Delta_{ij},\Delta_{kl}}(1-\xi)\>.$
(2.17)

With the help of these results, the process of writing all crossing equations
becomes straightforward. Given a set of external operators, one simply lists
all non-vanishing four-point functions. Then, for each inequivalent ordering,
eq. (2.16) gives the relevant crossing equations.

####  2.3.1 Comment on complex notation

In this work we consider scalar operators charged under $O(2)$ symmetry, so in
order to write the crossing equations we should take global-symmetry tensor
structures into account. Alternatively, we can exploit complex notation, which
means that for an $O(2)$ vector instead of working with a real field
$\phi^{i}$ with a two-valued index $i=1,2$, we work with a single complex
field $\phi=\phi^{1}+i\phi^{2}$ and its complex conjugate
$\bar{\phi}\equiv\phi^{*}$. In a completely analogous way, we can construct
operators with arbitrary charge. The advantage of complex notation is that it
eliminates the need to keep track of global-symmetry indices and tensor
structures. Using again the example of a vector under $O(2)$, the OPE
$\phi^{i}\times\phi^{j}$ contains the singlet representation ($S$), the
antisymmetric representation ($A$), and the symmetric-traceless representation
($T$), each with an associated tensor structure. Instead, in complex notation
we have the OPE $\phi\times\bar{\phi}$ with ${\mathcal{S}}$-parity even
operators corresponding to ($S$), the OPE $\phi\times\bar{\phi}$ with
${\mathcal{S}}$-parity odd operators corresponding to ($A$), and the OPE
$\phi\times\phi$ with ${\mathcal{S}}$-parity even operators corresponding to
($T$).

####  2.3.2 One complex scalar

The simplest case of crossing that we can consider is for one complex scalar
$\phi$. From the defect CFT perspective, this setup has three applications
depending on the interpretation given to $\phi$. In the first application,
which we consider in section 4.1.1, we take $\Delta_{\phi}=2$ and interpret
$\phi=\text{D}^{1}+i\text{D}^{2}\equiv\text{D}$ as the displacement operator
of a co-dimension two defect. In the second application, which we consider in
section 4.1.2, we take $\Delta_{\phi}=1$ and interpret
$\phi=t^{1}+it^{2}\equiv t$ as the tilt operator. In the third application,
which we consider in section 3.1, we can think of $\phi=\psi_{n+v}\>$ with
$n\in\mathbb{Z}$ as a defect mode in a monodromy defect.

Regardless of the interpretation of $\phi$, we can use eq. (2.16) on the
orderings $\langle\phi\bar{\phi}\phi\bar{\phi}\rangle$ and
$\langle\phi\phi\bar{\phi}\bar{\phi}\rangle$. We find a system of three
crossing equations which in vector notation read [47]

$\displaystyle\sum_{{\mathcal{O}}^{\pm}}(\lambda_{\phi\bar{\phi}{\mathcal{O}}})^{2}\vec{V}^{\phi\bar{\phi}}_{\Delta,S}+\sum_{{\mathcal{O}}^{+}}|\lambda_{\phi\phi{\mathcal{O}}}|^{2}\vec{V}^{\phi\phi}_{\Delta}=0\,,$
(2.18)

where the crossing vectors are given explicitly in eq. (B.1). The leftmost sum
runs over defect primaries of both $\mathcal{S}$-parities, while the rightmost
one include only $\mathcal{S}$-parity even defect primaries. The contribution
of the identity operator has not been separated explicitly, but is given by
$\vec{V}_{0,0}^{\phi\bar{\phi}}$.

##### One complex generalized free field.

A notable solution to the crossing equations in eq. (2.18) is complex
generalized free field theory, based on a complex scalar field $\phi$ with
scaling dimension $\Delta_{\phi}$, and its complex conjugate $\bar{\phi}$.
Using Wick’s theorem, the four-point function in the notation of eq. (2.12)
reads

$\displaystyle G_{\phi\bar{\phi}\phi\bar{\phi}}(\xi)$
$\displaystyle=1+\left(\frac{\xi}{1-\xi}\right)^{2\Delta_{\phi}}=1+\sum_{p=0}^{\infty}c_{p}(\Delta_{\phi},\Delta_{\phi})g_{2\Delta_{\phi}+p}^{0,0}(\xi)\>,$
(2.19) $\displaystyle G_{\phi\phi\bar{\phi}\bar{\phi}}(\xi)$
$\displaystyle=\alpha\xi^{2\Delta_{\phi}}+\left(\frac{\xi}{1-\xi}\right)^{2\Delta_{\phi}}=\sum_{p=0}^{\infty}\big{(}1+\alpha(-1)^{p}\big{)}c_{p}(\Delta_{\phi},\Delta_{\phi})g_{2\Delta_{\phi}+p}^{0,0}(\xi)~{},$
(2.20)

with [48, 25]

$\displaystyle
c_{p}(\Delta_{1},\Delta_{2})=\frac{(2\Delta_{1})_{p}(2\Delta_{2})_{p}}{p!(2\Delta_{1}+2\Delta_{2}+p-1)_{p}}\>.$
(2.21)

The second correlator above contains a parameter $\alpha$, such that
$-1\leq\alpha\leq 1$ in unitary theories. For $\alpha=1$ the solution
corresponds to a generalized free boson (GFB), while for $\alpha=-1$ it
corresponds to a generalized free fermion (GFF).

It was noted in [25] that the solution to the crossing equation of the
generalized real free fermion
$\langle\psi(\tau_{1})\psi(\tau_{2})\psi(\tau_{3})\psi(\tau_{4})\rangle$ can
be extended to a solution of the crossing equations in eq. (2.18), if the gap
on the scaling dimension of the first operator in the traceless-symmetric
representation does not exceed $2\Delta_{\psi}$. The spectra for the
${\mathcal{S}}$-parity even and odd operators are equal in this case. For the
displacement operator, this happens for $\Delta_{{\text{D}^{2}}}<2\times 2=4$,
while for the tilt operator the solution appears for $\Delta_{t^{2}}<2\times
1=2$.

####  2.3.3 Tilt and displacement

Next we consider a mixed system between the tilt operator $t$ and the
displacement operator D, where we again use complex notation. In order to
write down the crossing equations we note that the $\mathcal{S}$-parity even
singlets can appear both in $t\times\bar{t}$ and
$\text{D}\times\bar{\text{D}}$, while the $\mathcal{S}$-parity odd channels
are different in the two OPEs, because the tilt and displacement transform
under different $O(2)$ groups. All in all, crossing reads

$\displaystyle\sum_{{\mathcal{O}}^{+}}(\lambda_{t\bar{t}{\mathcal{O}}}$
$\displaystyle\;\lambda_{\text{D}\bar{\text{D}}{\mathcal{O}}})\vec{V}^{+}_{\Delta}\begin{pmatrix}\lambda_{t\bar{t}{\mathcal{O}}}\\\
\lambda_{\text{D}\bar{\text{D}}{\mathcal{O}}}\end{pmatrix}+\sum_{{\mathcal{O}}^{-}}|\lambda_{t\bar{t}{\mathcal{O}}}|^{2}\vec{V}^{t\bar{t},-}_{\Delta}+\sum_{{\mathcal{O}}^{-}}|\lambda_{\text{D}\bar{\text{D}}{\mathcal{O}}}|^{2}\vec{V}^{\text{D}\bar{\text{D}},-}_{\Delta}$
$\displaystyle+\sum_{{\mathcal{O}}^{+}}|\lambda_{tt{\mathcal{O}}}|^{2}\vec{V}^{tt}_{\Delta}+\sum_{{\mathcal{O}}^{+}}|\lambda_{\text{D}\text{D}{\mathcal{O}}}|^{2}\vec{V}^{\text{D}\text{D}}_{\Delta}+\sum_{{\mathcal{O}}^{\pm}}|\lambda_{t\text{D}{\mathcal{O}}}|^{2}\vec{V}^{t\text{D}}_{\Delta,S}=0\>.$
(2.22)

Once again the crossing vectors are presented in appendix B.2.

####  2.3.4 One real scalar and the tilt

The third setup we study is crossing for one real scalar $\phi_{1}$ and a tilt
operator $t$, for which we use complex notation. This has applications to the
magnetic line defect of section 3.2, where $\phi_{1}$ corresponds to the
scalar that breaks the bulk symmetry, and the tilt operator must be present
due to the symmetry breaking. The numerical results for this setup will be
discussed in section 4.3. The crossing equations can be obtain in a similar
way as before, but we now obtain seven independent equations that in vector
notation read

$\displaystyle\sum_{{\mathcal{O}}^{+}}$
$\displaystyle(\lambda_{\phi_{1}\phi_{1}{\mathcal{O}}}\;\lambda_{t\bar{t}{\mathcal{O}}})\vec{V}^{+}_{\Delta}\begin{pmatrix}\lambda_{\phi_{1}\phi_{1}{\mathcal{O}}}\\\
\lambda_{t\bar{t}{\mathcal{O}}}\end{pmatrix}+\sum_{{\mathcal{O}}^{-}}(\lambda_{t\bar{t}{\mathcal{O}}})^{2}\vec{V}^{-}_{\Delta}+\sum_{{\mathcal{O}}^{+}}|\lambda_{tt{\mathcal{O}}}|^{2}\vec{V}^{tt}_{\Delta}+\sum_{{\mathcal{O}}^{\pm}}|\lambda_{\phi_{1}t{\mathcal{O}}}|^{2}\vec{V}^{\phi_{1}t}_{\Delta,S}=0\>.$
(2.23)

The crossing vectors are found in appendix B.3. _Mutatis mutandis_ , the same
system of crossing equations can be used to study the mixed correlators of
$\phi_{1}$ and D. We will leave this idea for future exploration.

####  2.3.5 Two complex scalars

Finally we study the crossing equations of two unequal complex scalars
$\phi_{1}$ and $\phi_{2}$. We apply this to monodromy defects, see section
4.2, where $\phi_{1}=\psi_{n_{1}+v}$ and $\phi_{2}=\psi_{n_{2}+v}$ are two
different defect modes of a bulk scalar field. In total there are twelve
crossing equations and five different OPE channels [49]

$\displaystyle\sum_{{\mathcal{O}}^{\pm}}(\lambda_{\phi_{1}\bar{\phi}_{1}{\mathcal{O}}}$
$\displaystyle\;\lambda_{\phi_{2}\bar{\phi}_{2}{\mathcal{O}}})\vec{V}_{\Delta,S}\begin{pmatrix}\lambda_{\phi_{1}\bar{\phi}_{1}{\mathcal{O}}}\\\
\lambda_{\phi_{2}\bar{\phi}_{2}{\mathcal{O}}}\end{pmatrix}+\sum_{{\mathcal{O}}^{+}}|\lambda_{\phi_{1}\phi_{1}{\mathcal{O}}}|^{2}\vec{V}^{11}_{\Delta}+\sum_{{\mathcal{O}}^{+}}|\lambda_{\phi_{2}\phi_{2}{\mathcal{O}}}|^{2}\vec{V}^{22}_{\Delta}$
$\displaystyle+\sum_{{\mathcal{O}}^{\pm}}|\lambda_{\phi_{1}\phi_{2}{\mathcal{O}}}|^{2}\vec{V}^{12}_{\Delta,S}+\sum_{{\mathcal{O}}^{\pm}}|\lambda_{\phi_{1}\bar{\phi}_{2}{\mathcal{O}}}|^{2}\vec{V}^{1\bar{2}}_{\Delta,S}=0\>.$
(2.24)

The crossing vectors can be found in appendix B.4.

##### Two complex generalized free fields.

The OPE coefficients that contain only $\phi_{1}$ or only $\phi_{2}$ follow
from our discussion above. The new information is contained in the
$\phi_{1}\times\phi_{2}$ OPE, which can be analyzed from the following four-
point function

$\displaystyle\begin{split}G_{\phi_{1}\phi_{2}\bar{\phi}_{2}\bar{\phi}_{1}}(\xi)&=\frac{\xi^{\Delta_{1}+\Delta_{2}}}{(1-\xi)^{2\Delta_{2}}}=\sum_{p=0}^{\infty}c_{p}(\Delta_{1},\Delta_{2})g_{\Delta_{1}+\Delta_{2}+p}^{\Delta_{12},\Delta_{21}}(\xi)\>.\end{split}$
(2.25)

From here we immediately read off
$|\lambda_{\phi_{1}\phi_{2}{\mathcal{O}}}|^{2}$, and by sending
$\phi_{2}\to\bar{\phi}_{2}$ we find that the same formula applies to
$|\lambda_{\phi_{1}\bar{\phi}_{2}{\mathcal{O}}}|^{2}$.

##  3 Defect theories in the $\varepsilon$-expansion

In this section we use perturbation theory to study two important examples of
conformal line defects: the $SO(2)_{F}$-preserving monodromy defect and the
$O(3)$-breaking magnetic line defect. We start by reviewing known results on
the $SO(2)_{F}$ monodromy defect in the $\varepsilon$-expansion, which has
been studied in great detail in [38, 39, 40]. In view of the comparison to the
numerical bootstrap results, we add a few OPE coefficients to the CFT data
already available in the literature, which can be straightforwardly extracted
from the results of [39]. Then, we study the magnetic line defect in the
$\varepsilon$-expansion, and present new results which complement the study
performed in [5]. We compare these perturbative results with the predictions
from the numerical conformal bootstrap in section 4.

###  3.1 Monodromy defects with $SO(2)_{F}$ symmetry

##### Monodromy defects in the free $O(2)$ model.

The first example we discuss is a $U(1)_{F}$-preserving monodromy defect in
free theory. This defect, first considered in [38, 39, 40], generalizes the
$\mathbb{Z}_{2}$ twist defect defined in [2, 25]. Following [38], we start
from a set of $N=2$ free real scalars $\phi^{i}$ in the bulk that satisfy

$\phi^{i}(r,\theta+2\pi,\vec{x})=g^{ij}\phi^{j}(r,\theta,\vec{x})\>,\quad
g^{ij}\in O(2)_{F}\>.$ (3.1)

The scalars $\phi^{i}$ either get mixed into each other, obtain a minus sign,
or remain unchanged when going around the monodromy defect. In terms of the
complex combination

$\Phi=\phi^{1}+i\phi^{2}\>,$ (3.2)

the most general $U(1)_{F}$ monodromy becomes101010As explained in [40] this
monodromy can be thought of as a large and constant $U(1)$ background gauge
transformation for $\Phi$.

$\Phi(r,\theta+2\pi,\vec{x})=e^{2\pi iv}\Phi(r,\theta,\vec{x})\>,\quad v\sim
v+1~{},\quad v\in[0,1)~{}.$ (3.3)

The complex scalar $\Phi$ has an internal $U(1)_{F}$ symmetry that gets
enhanced to $O(2)_{F}$ for $v=0$ (the trivial defect) and for $v=\frac{1}{2}$
(the $\mathbb{Z}_{2}$ monodromy defect). For these values of $v$, the
transformation $\Phi\to\bar{\Phi}$, which belongs to $O(2)_{F}$ but not
$U(1)\simeq SO(2)_{F}$, is a symmetry. As a consequence of the monodromy, the
local primary operators allowed to appear in the bulk-defect expansion of
$\Phi$ will generically have non-integer spin $s\in\mathbb{Z}+v$, i.e.

$\displaystyle\Phi(r,\theta,\vec{x})\underset{r\rightarrow
0}{\sim}\sum_{\Psi_{s}}\sum_{s\in\mathbb{Z}+v}\frac{e^{-i\theta
s}}{r^{\Delta_{\Phi}-\Delta_{\Psi_{s}}}}\Psi_{s}(\vec{x})+\text{c.c.}$ (3.4)

The scaling dimensions of the defect modes of $\Phi$ are completely fixed as a
consequence of the bulk free equation of motion, and read (see e.g. [2, 25,
41])

$\displaystyle\Delta_{\Psi_{s}}=\Delta_{\Phi}+|s|=\frac{d-2}{2}+|s|\,.$ (3.5)

In terms of real bulk scalar
$\phi^{i}\sim\sum_{s}e^{is\theta}\psi_{s}^{i}+\text{c.c.}$, the reality
condition is that $\bar{\psi}_{s}^{i}=\psi_{-s}^{i}$ and so
$\Psi_{s}=\psi^{1}_{s}+i\psi^{2}_{s}$ satisfies
$\bar{\Psi}_{s}\equiv\bar{\psi}^{1}_{s}-i\bar{\psi}^{2}_{s}=\psi^{1}_{-s}-i\psi^{2}_{-s}$.

##### Monodromy defects in the interacting $O(2)$ model.

The simplest example of an interacting $U(1)_{F}$ monodromy defect is obtained
by imposing the condition of eq. (3.3) on the fundamental vector of the
critical 3d $O(2)$ vector model. In perturbation theory this example is
tractable in the standard framework of $\varepsilon$-expansion (with fixed co-
dimension $q=2$). The bulk is tuned to the Wilson-Fisher fixed point with
coupling111111There is a slight difference in notation with respect to [39],
where the symmetry group in the bulk and on the defect is $O(2N)$, while we
will denote it by $O(N)$ with $N=$ even, and in particular take $N=2$, to
avoid differences in notation throughout the paper.

$\lambda^{*}=\frac{8\pi^{2}}{10}+O(\varepsilon^{2})\>.$ (3.6)

The scaling dimensions of the defect modes $\Psi_{s}$ are found to be [38, 39]

$\Delta_{\Psi_{s}}=1+|s|-\frac{\varepsilon}{2}+\frac{1}{5}\frac{v(v-1)}{|s|}\varepsilon+O(\varepsilon^{2})\>.$
(3.7)

For the monodromy defect, the displacement operator D appears in the OPE of
two defect modes with spins $|s|=v$ and $|s|=v-1$:

$\displaystyle\Psi_{v}\times\bar{\Psi}_{v-1}\sim\text{D}~{},\quad\bar{\Psi}_{v}\times{\Psi}_{v-1}\sim{\bar{\text{D}}}~{}.$
(3.8)

In appendix A we show that it appears in the $\Psi_{v}\times\bar{\Psi}_{v-1}$
OPE with (squared) OPE coefficient

$|\lambda_{\Psi_{v}\bar{\Psi}_{v-1}\text{D}}|^{2}=1+\frac{\varepsilon}{10}\left(2H^{1-v}+2H^{v}-3\right)\>,$
(3.9)

where $H^{v}$ is the analytic continuation of the harmonic number. We will
also need the scaling dimensions and correponding OPE coefficients of the
leading singlets in

$\displaystyle\Psi_{v}\times\bar{\Psi}_{v}\sim{\mathds{1}}+\mathcal{O}_{0}+\dots,\quad\Psi_{v-1}\times\bar{\Psi}_{v-1}\sim{\mathds{1}}+\mathcal{O}_{0}+\dots.$
(3.10)

In appendix A we show that

$\displaystyle\Delta_{\mathcal{O}_{0}}=2+2v-\varepsilon+\varepsilon\left(\frac{4}{5(1+2v)}+\frac{2(v-1)}{5}\right)~{}.$
(3.11)

In section 4.2 we will compare the $\varepsilon$-expansion predictions to the
numerical bootstrap results.

It would be interesting to compute the correlator
$\langle\text{D}\text{D}{\bar{\text{D}}}{\bar{\text{D}}}\rangle$ at the first
non-trivial order in $\varepsilon$-expansion to compare this to our single-
correlator numerical bounds for the displacement operator. We will leave this
for future work.

###  3.2 Localized magnetic field line defect

Let us continue with the determination of three-point OPE coefficients of
defect operators for the magnetic line defect. The calculation is based on and
extends the work [5], which focused on the scaling dimensions of low-lying
defect operators. We obtain these results from a Feynman diagram expansion,
keeping terms up to order $O(\varepsilon)$ in the $\varepsilon$-expansion. In
order to obtain properly normalized OPE coefficients, we need to determine
both two- and three-point functions. At the end, we also calculate several
four-point functions, that by means of the conformal block decomposition allow
us to check our results and obtain further coefficients.

####  3.2.1 Overview

To study the magnetic line defect we use a Lagrangian description that couples
the $O(N)$ model to a magnetic field localized along an infinite line. Using
rotation invariance we take the line to be oriented as
$x^{\mu}(\tau)=(\tau,\vec{0})$, while using $O(N)$ invariance, we choose the
magnetic field to be in the $\phi_{1}$ direction. All in all, the action is

$\displaystyle S=\int
d^{d}x\left(\frac{1}{2}(\partial_{\mu}\phi_{a})^{2}+\frac{\lambda_{0}}{4!}(\phi_{a}^{2})^{2}\right)+h_{0}\int_{-\infty}^{\infty}d\tau\,\phi_{1}(x(\tau))\,,\qquad
a=1,\ldots,N\,.$ (3.12)

Let us start reviewing the results of [5], which motivate our bootstrap setup
in section 4.3. Because the defect in eq. (3.12) breaks the global symmetry
$O(N)\to O(N-1)$, there exists a tilt operator besides the displacement
operator. In perturbation theory, the tilt and displacement operators are
identified as

$\displaystyle
t_{\hat{a}}\propto\phi_{\hat{a}}\,,\qquad\text{D}\propto\nabla\phi_{1}\,,\qquad\hat{a}=2,\ldots,N\,,$
(3.13)

and as usual they have protected dimension $\Delta_{t}=1$ and
$\Delta_{\text{D}}=2$. After the tilt, the operator with the second-lowest
dimension is the localized magnetic field $\phi_{1}$. The scaling dimension of
$\phi_{1}$ to two-loop order in the $\varepsilon$-expansion reads

$\displaystyle\Delta_{\phi_{1}}=1+\varepsilon-\frac{3N^{2}+49N+194}{2(N+8)^{2}}\varepsilon^{2}+O(\varepsilon^{3})\;\;\xrightarrow{\text{Pad\'{e}}}\;1.55\,.$
(3.14)

Furthermore, inputting information from $d=2$ in a Padé approximant, the
authors of [5] estimated the value $\Delta_{\phi_{1}}\approx 1.55$, which is
also consistent with their $1/N$ results and Monte-Carlo simulations [50, 51].

Since the two lowest-dimensional operators on the defect are $t_{\hat{a}}$ and
$\phi_{1}$, a natural candidate for a bootstrap study is the mixed correlator
involving them, which is the one-dimensional analog of [52, 53]. In order to
motivate gap assumptions in the numerical study, let us look at the lowest-
lying operators in the different OPE channels. In the singlet channel $(S)$,
which appears for $\phi_{1}\times\phi_{1}$ or $(t_{\hat{a}}\times
t_{\hat{b}})_{S}$, the lowest dimension operators are
$\phi_{1}+s_{-}+s_{+}+\ldots$. Here $s_{\pm}$ are linear combinations of
$\phi_{1}^{2}$ and $\phi_{a}^{2}$ with scaling dimension

$\displaystyle\Delta_{s_{\pm}}$
$\displaystyle=2+\varepsilon\,\frac{3N+20\pm\sqrt{N^{2}+40N+320}}{2(N+8)}+O(\varepsilon^{2})\,.$
(3.15)

Similarly, vector operators appear in the OPE $\phi_{1}\times
t_{\hat{a}}=t_{\hat{a}}+V_{\hat{a}}+\ldots$. The leading vector is
$V_{\hat{a}}\propto\phi_{1}\phi_{\hat{a}}$, and its dimension reads

$\displaystyle\Delta_{V}$
$\displaystyle=2+\varepsilon\,\frac{N+10}{N+8}+O(\varepsilon^{2})\>.$ (3.16)

Finally, in the OPE $t_{\hat{a}}\times t_{\hat{b}}$ there is an antisymmetric
channel $(A)$, where the lowest-dimensional operator is
$A_{\hat{a}\hat{b}}\propto\phi_{[\hat{a}}\phi_{\hat{b}]}$, and a symmetric-
traceless channel $(T)$ with the lowest-lying operator
$T_{\hat{a}\hat{b}}\propto\phi_{\hat{a}}\phi_{\hat{b}}$. Their dimensions are
given by

$\displaystyle\Delta_{A}=3+O(\varepsilon^{2})\,,\qquad\Delta_{T}=2+\frac{2\varepsilon}{N+8}+O(\varepsilon^{2})\>.$
(3.17)

Besides scaling dimensions, the numerical conformal bootstrap can also probe
the three-point OPE coefficients of these operators. The goal of the rest of
the section will be to compute these OPE coefficients to leading order in the
$\varepsilon$-expansion, see table 1 for a summary of the main results.
Besides two- and three-point functions, we also compute several four-point
functions, which thanks to the conformal block decomposition, contain
information of many other OPE coefficients.

$\lambda_{\phi_{1}\phi_{1}\phi_{1}}$ |  $\lambda_{tt\phi_{1}}$ |  $\lambda_{\phi_{1}\phi_{1}s_{\pm}}$ |  $\lambda_{tts_{\pm}}$ |  $\lambda_{\phi_{1}tV}$ |  $\lambda_{ttA}$ |  $\lambda_{ttT}$
---|---|---|---|---|---|---
(3.50) |  (3.50) |  (3.55) |  (3.56) |  (3.57) |  (3.62) |  (3.57)
Table 1: Summary of the most important OPE coefficients computed in this
section. Further coefficients appear in (3.52) or can be extracted from the
four-point functions in section 3.2.5.

####  3.2.2 Conventions

Throughout this section we follow the conventions of [5], so in particular the
action is given by eq. (3.12) and the scalar propagator in free theory is

$\displaystyle\leavevmode\hbox to23.16pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\,\equiv\;\langle\phi_{a}(x_{1})\phi_{b}(x_{2})\rangle_{\lambda_{0}=h_{0}=0}\,=\;\frac{\kappa\delta_{ab}}{(x_{12}^{2})^{1-\frac{\varepsilon}{2}}}\,,\qquad\kappa=\frac{\Gamma\\!\left(\frac{d}{2}\right)}{2\pi^{d/2}(d-2)}\,.$
(3.18)

In perturbative expansions there is a bulk four-point vertex and a vertex that
couples a bulk operator to the defect

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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}{}\pgfsys@moveto{-5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{-5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;\equiv\;-\lambda_{0}\int
d^{d}x\ldots\,,\qquad\leavevmode\hbox to23.56pt{\vbox
to18.89pt{\pgfpicture\makeatletter\hbox{\hskip 3.24544pt\lower-1.6143pt\hbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@lineto{8.5359pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{9.95021pt}{0.0pt}\pgfsys@curveto{9.95021pt}{0.7811pt}{9.317pt}{1.4143pt}{8.5359pt}{1.4143pt}\pgfsys@curveto{7.7548pt}{1.4143pt}{7.1216pt}{0.7811pt}{7.1216pt}{0.0pt}\pgfsys@curveto{7.1216pt}{-0.7811pt}{7.7548pt}{-1.4143pt}{8.5359pt}{-1.4143pt}\pgfsys@curveto{9.317pt}{-1.4143pt}{9.95021pt}{-0.7811pt}{9.95021pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.5359pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;\equiv\;-h_{0}\int_{-\infty}^{\infty}d\tau\ldots\,.$
(3.19)

Notice that only $\phi_{1}$ couples to the line, and not all $\phi_{\hat{a}}$
for $\hat{a}=2,\ldots,N$. We work in dimensional regularization with minimal
subtraction, so the bare couplings $\lambda_{0}$ and $h_{0}$ are related to
the renormalized ones as

$\displaystyle\lambda_{0}=\lambda
M^{\varepsilon}\left(1+\frac{\lambda}{(4\pi)^{2}}\frac{N+8}{3\varepsilon}+O(\lambda^{2})\right)\,,\quad
h_{0}=hM^{\varepsilon/2}\left(1+\frac{\lambda}{(4\pi)^{2}}\frac{h^{2}}{12\varepsilon}+O(\lambda^{2})\right)\,.$
(3.20)

The renormalized couplings depend on the renormalization scale $M$ as

$\displaystyle\beta_{\lambda}$
$\displaystyle=M\frac{d\lambda}{dM}=-\lambda\varepsilon+\frac{\lambda^{2}}{(4\pi)^{2}}\frac{N+8}{3}+O(\lambda^{3})\,,$
$\displaystyle\beta_{h}$
$\displaystyle=M\frac{dh}{dM}=-\frac{h\varepsilon}{2}+\frac{\lambda}{(4\pi)^{2}}\frac{h^{3}}{6}+O(\lambda^{2})\,,$
(3.21)

so there exists a non-trivial fixed point where we evaluate most of our
results:

$\displaystyle\frac{\lambda_{*}}{(4\pi)^{2}}=\frac{3\varepsilon}{N+8}+O(\varepsilon^{2})\,,\qquad
h_{*}^{2}=N+8+O(\varepsilon)\,.$ (3.22)

Since we are interested in obtaining OPE coefficients at order
$O(\varepsilon)$, we shall consider diagrams with at most one bulk vertex
insertion $\lambda_{*}\sim O(\varepsilon)$. On the other hand, we have to
allow an arbitrary number of defect insertions because $h_{*}\sim O(1)$.
However, in practice only a finite number of diagrams will contribute at any
given order in $\varepsilon$.

We often split the coordinates into a direction $\tau$ parallel to the defect
and $\vec{x}\in\mathbb{R}^{d-1}$ directions orthogonal to the defect. We shall
only consider correlation functions of operators that live on the defect, for
which we use the shorthand notation
${\mathcal{O}}(\tau)={\mathcal{O}}(\tau,\vec{x}=0)$.

####  3.2.3 Two-point functions

We obtain all two-point functions of operators of the schematic form
$\phi,\,\phi^{2},\,\nabla\phi$, where $\phi$ is the fundamental scalar and
$\nabla$ are derivatives orthogonal to the defect. Besides rederiving their
scaling dimensions, which appeared previously in [5], we obtain the overall
normalization, which is needed in the calculation of higher-point functions.

#### The correlator $\langle\,\phi\,\phi\,\rangle$

Let us start with the simplest example: the two-point function of the
fundamental field. To order $O(\varepsilon)$ only two diagrams contribute:

$\displaystyle\langle\,\phi_{a}(\tau)\,\phi_{b}(0)\,\rangle\;=\;\leavevmode\hbox
to40.63pt{\vbox to11.7pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{11.09575pt}{34.14322pt}{11.09575pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;+\;\leavevmode\hbox
to40.63pt{\vbox to12.91pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-1.6143pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{11.09575pt}{34.14322pt}{11.09575pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{19.91684pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@lineto{19.91684pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33115pt}{7.96674pt}\pgfsys@curveto{21.33115pt}{8.74783pt}{20.69794pt}{9.38104pt}{19.91684pt}{9.38104pt}\pgfsys@curveto{19.13574pt}{9.38104pt}{18.50253pt}{8.74783pt}{18.50253pt}{7.96674pt}\pgfsys@curveto{18.50253pt}{7.18564pt}{19.13574pt}{6.55243pt}{19.91684pt}{6.55243pt}\pgfsys@curveto{20.69794pt}{6.55243pt}{21.33115pt}{7.18564pt}{21.33115pt}{7.96674pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{7.96674pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.91684pt}{7.96674pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{15.64069pt}{0.0pt}\pgfsys@curveto{15.64069pt}{0.7811pt}{15.00748pt}{1.4143pt}{14.22638pt}{1.4143pt}\pgfsys@curveto{13.44528pt}{1.4143pt}{12.81207pt}{0.7811pt}{12.81207pt}{0.0pt}\pgfsys@curveto{12.81207pt}{-0.7811pt}{13.44528pt}{-1.4143pt}{14.22638pt}{-1.4143pt}\pgfsys@curveto{15.00748pt}{-1.4143pt}{15.64069pt}{-0.7811pt}{15.64069pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.22638pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{27.0216pt}{0.0pt}\pgfsys@curveto{27.0216pt}{0.7811pt}{26.3884pt}{1.4143pt}{25.6073pt}{1.4143pt}\pgfsys@curveto{24.8262pt}{1.4143pt}{24.193pt}{0.7811pt}{24.193pt}{0.0pt}\pgfsys@curveto{24.193pt}{-0.7811pt}{24.8262pt}{-1.4143pt}{25.6073pt}{-1.4143pt}\pgfsys@curveto{26.3884pt}{-1.4143pt}{27.0216pt}{-0.7811pt}{27.0216pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.6073pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;+\;\ldots\,.$ (3.23)

The first diagram is obtained setting $\vec{x}_{1}=\vec{x}_{2}=0$ in the free
propagator (3.18). For the integral that enters the second diagram, we first
integrate the two defect insertions using

$\displaystyle\int_{-\infty}^{\infty}\frac{d\tau}{(|\vec{x}|^{2}+\tau^{2})^{\Delta}}=\frac{\sqrt{\pi}\,\Gamma\\!\left(\Delta-\frac{1}{2}\right)}{\Gamma(\Delta)}\frac{1}{|\vec{x}^{2}|^{\Delta-\frac{1}{2}}}\,,$
(3.24)

and then the bulk vertex using

$\displaystyle\int
d^{d}x_{3}\,\frac{|\vec{x}_{3}|^{\varepsilon}}{(x_{13}^{2}x_{23}^{2})^{1-\frac{\varepsilon}{2}}}=\frac{\sqrt{\pi}\,\Gamma\\!\left(1-\frac{3\varepsilon}{2}\right)\Gamma(\varepsilon)^{2}\,\Gamma\\!\left(\frac{\varepsilon+1}{2}\right)}{2\Gamma\\!\left(1-\frac{\varepsilon}{2}\right)^{2}\Gamma(2\varepsilon)}\frac{1}{|\tau_{12}^{2}|^{1-\frac{3\varepsilon}{2}}}\,,$
(3.25)

which follows from Schwinger parametrization. Using these results, it is a
simple bookkeeping exercise to obtain the contribution of each diagram:

$\displaystyle\leavevmode\hbox to40.63pt{\vbox
to11.7pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-0.4pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{11.09575pt}{34.14322pt}{11.09575pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{\kappa\delta_{ab}}{\tau^{2-\varepsilon}}\,,$
$\displaystyle\leavevmode\hbox to40.63pt{\vbox
to12.91pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-1.6143pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{11.09575pt}{34.14322pt}{11.09575pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{19.91684pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@lineto{19.91684pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33115pt}{7.96674pt}\pgfsys@curveto{21.33115pt}{8.74783pt}{20.69794pt}{9.38104pt}{19.91684pt}{9.38104pt}\pgfsys@curveto{19.13574pt}{9.38104pt}{18.50253pt}{8.74783pt}{18.50253pt}{7.96674pt}\pgfsys@curveto{18.50253pt}{7.18564pt}{19.13574pt}{6.55243pt}{19.91684pt}{6.55243pt}\pgfsys@curveto{20.69794pt}{6.55243pt}{21.33115pt}{7.18564pt}{21.33115pt}{7.96674pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{7.96674pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.91684pt}{7.96674pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{15.64069pt}{0.0pt}\pgfsys@curveto{15.64069pt}{0.7811pt}{15.00748pt}{1.4143pt}{14.22638pt}{1.4143pt}\pgfsys@curveto{13.44528pt}{1.4143pt}{12.81207pt}{0.7811pt}{12.81207pt}{0.0pt}\pgfsys@curveto{12.81207pt}{-0.7811pt}{13.44528pt}{-1.4143pt}{14.22638pt}{-1.4143pt}\pgfsys@curveto{15.00748pt}{-1.4143pt}{15.64069pt}{-0.7811pt}{15.64069pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.22638pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{27.0216pt}{0.0pt}\pgfsys@curveto{27.0216pt}{0.7811pt}{26.3884pt}{1.4143pt}{25.6073pt}{1.4143pt}\pgfsys@curveto{24.8262pt}{1.4143pt}{24.193pt}{0.7811pt}{24.193pt}{0.0pt}\pgfsys@curveto{24.193pt}{-0.7811pt}{24.8262pt}{-1.4143pt}{25.6073pt}{-1.4143pt}\pgfsys@curveto{26.3884pt}{-1.4143pt}{27.0216pt}{-0.7811pt}{27.0216pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.6073pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=-\frac{\lambda_{0}h_{0}^{2}}{(4\pi)^{4}}\,\frac{2\delta_{1a}\delta_{1b}+\delta_{ab}}{\tau^{2-3\varepsilon}}\left(\frac{2}{3\varepsilon}-\frac{1}{3}+\aleph+O(\varepsilon)\right)\,.$
(3.26)

To simplify the notation, we have introduced the constant

$\displaystyle\aleph\equiv 1+\gamma_{E}+\log\pi\,,$ (3.27)

which appears repeatedly in the calculations below but drops out of physical
quantities such as scaling dimensions and OPE coefficients. We now introduce
renormalized fields, which transform irreducibly under the unbroken symmetry
group $O(N-1)$:

$\displaystyle\phi_{1}\equiv
Z_{\phi_{1}}[\phi_{1}]\,,\qquad\phi_{\hat{a}}\equiv Z_{t}\,t_{\hat{a}}\,.$
(3.28)

The operator $t_{\hat{a}}$ is the tilt operator from section 2.2.2. The two
renormalization factors are obtained demanding that poles in $\varepsilon$
cancel in (3.23), giving

$\displaystyle
Z_{\phi_{1}}=1-\frac{\lambda}{(4\pi)^{2}}\frac{h^{2}}{4\varepsilon}+O(\lambda^{2})\,,\qquad
Z_{t}=1-\frac{\lambda}{(4\pi)^{2}}\frac{h^{2}}{12\varepsilon}+O(\lambda^{2})\,.$
(3.29)

From the renormalization factor one can immediately obtain the anomalous
dimension at the critical point using $\gamma_{\mathcal{O}}=M\frac{d}{dM}\log
Z_{\mathcal{O}}$. The results have been summarized in section 3.2.1, and are
in perfect agreement with [5]. Finally, note that the two-point functions at
the critical point read

$\displaystyle\langle\,[\phi_{1}](\tau)\,[\phi_{1}](0)\,\rangle$
$\displaystyle=\frac{{\mathcal{N}}_{\phi_{1}}^{2}}{\tau^{2\Delta_{\phi_{1}}}}\,,\quad\quad\;{\mathcal{N}}_{\phi_{1}}^{2}=\kappa\left(1-\frac{3\aleph}{2}\,\varepsilon+O(\varepsilon^{2})\right),$
$\displaystyle\langle\,t_{\hat{a}}(\tau)\,t_{\hat{b}}(0)\,\rangle$
$\displaystyle=\delta_{\hat{a}\hat{b}}\,\frac{{\mathcal{N}}_{t}^{2}}{\tau^{2}}\,,\quad\quad\;{\mathcal{N}}_{t}^{2}=\kappa\left(1-\frac{\aleph}{2}\,\varepsilon+O(\varepsilon^{2})\right).$
(3.30)

#### The correlator $\langle\,\nabla\phi\,\nabla\phi\,\rangle$

Similarly, we can consider the two-point function
$\langle\nabla_{i}\phi_{a}\nabla_{j}\phi_{b}\rangle$, which again consists of
two diagrams at this order:

$\displaystyle\leavevmode\hbox to40.63pt{\vbox
to11.7pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-0.4pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{11.09575pt}{34.14322pt}{11.09575pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\kappa(d-2)\frac{\delta_{ij}\delta_{ab}}{\tau^{4-\varepsilon}}\,,$
$\displaystyle\leavevmode\hbox to40.63pt{\vbox
to12.91pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-1.6143pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{21.33115pt}{7.96674pt}\pgfsys@curveto{21.33115pt}{8.74783pt}{20.69794pt}{9.38104pt}{19.91684pt}{9.38104pt}\pgfsys@curveto{19.13574pt}{9.38104pt}{18.50253pt}{8.74783pt}{18.50253pt}{7.96674pt}\pgfsys@curveto{18.50253pt}{7.18564pt}{19.13574pt}{6.55243pt}{19.91684pt}{6.55243pt}\pgfsys@curveto{20.69794pt}{6.55243pt}{21.33115pt}{7.18564pt}{21.33115pt}{7.96674pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{7.96674pt}\pgfsys@fillstroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.22638pt}{0.0pt}\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
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}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{27.0216pt}{0.0pt}\pgfsys@curveto{27.0216pt}{0.7811pt}{26.3884pt}{1.4143pt}{25.6073pt}{1.4143pt}\pgfsys@curveto{24.8262pt}{1.4143pt}{24.193pt}{0.7811pt}{24.193pt}{0.0pt}\pgfsys@curveto{24.193pt}{-0.7811pt}{24.8262pt}{-1.4143pt}{25.6073pt}{-1.4143pt}\pgfsys@curveto{26.3884pt}{-1.4143pt}{27.0216pt}{-0.7811pt}{27.0216pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.6073pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=-\frac{\lambda_{0}h_{0}^{2}}{(4\pi)^{4}}\,\frac{\delta_{ij}(2\delta_{1a}\delta_{1b}+\delta_{ab})}{\tau^{4-3\varepsilon}}\left(\frac{4}{9\varepsilon}-\frac{8}{27}+\frac{2}{3}\aleph+O(\varepsilon)\right)\,.$
(3.31)

The integrals are computed as before, but one needs to be careful to first
take transverse derivatives with respect to $\vec{x}_{1}$ and $\vec{x}_{2}$
and then sending $\vec{x}_{1},\vec{x}_{2}\to 0$. Once again we introduce
renormalized fields

$\displaystyle\nabla\phi_{1}\equiv
Z_{\text{D}}\,\text{D}\,,\qquad\nabla\phi_{\hat{a}}\equiv
Z_{\nabla\phi}\,[\nabla\phi_{\hat{a}}]\,,$ (3.32)

where D is the displacement operator which transforms as a vector under
$SO(d-1)$ transverse rotations. Cancelling poles in $\varepsilon$ we find

$\displaystyle
Z_{\text{D}}=1-\frac{\lambda}{(4\pi)^{2}}\frac{h^{2}}{12\varepsilon}+O(\lambda^{2})\,,\qquad
Z_{\nabla\phi}=1-\frac{\lambda}{(4\pi)^{2}}\frac{h^{2}}{36\varepsilon}+O(\lambda^{2})\,.$
(3.33)

It follows that the scaling dimension of the displacement is protected, and we
obtain the scaling dimension
$\Delta_{\nabla\phi}=2-\frac{1}{3}\varepsilon+O(\varepsilon^{2})$, in
agreement with [5]. The defect-defect two-point functions at the critical
point read

$\displaystyle\langle\,\text{D}_{i}(\tau)\,\text{D}_{j}(0)\,\rangle$
$\displaystyle=\delta_{ij}\,\frac{{\mathcal{N}}_{\text{D}}^{2}}{\tau^{4}}\,,\quad\quad\quad\,\;{\mathcal{N}}_{\text{D}}^{2}=2\kappa\left(1-\frac{4+3\aleph}{6}\,\varepsilon+O(\varepsilon^{2})\right),$
$\displaystyle\langle\,[\nabla_{i}\phi_{\hat{a}}](\tau)\,[\nabla_{j}\phi_{\hat{b}}](0)\,\rangle$
$\displaystyle=\delta_{\hat{a}\hat{b}}\delta_{ij}\,\frac{{\mathcal{N}}_{\nabla\phi}^{2}}{\tau^{2\Delta_{\nabla\phi}}}\,,\quad{\mathcal{N}}_{\nabla\phi}^{2}=2\kappa\left(1-\frac{10+3\aleph}{18}\,\varepsilon+O(\varepsilon^{2})\right).$
(3.34)

#### The correlator $\langle\,\phi^{2}\,\phi^{2}\,\rangle$

Finally, the last type of operators we are interested in are composites of two
fundamental fields:

$\displaystyle\langle\,\phi_{a}\phi_{b}(\tau)\,\phi_{c}\phi_{d}(\tau)\,\rangle\;\;=\;\;\leavevmode\hbox
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
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}{}\pgfsys@moveto{15.64069pt}{0.0pt}\pgfsys@curveto{15.64069pt}{0.7811pt}{15.00748pt}{1.4143pt}{14.22638pt}{1.4143pt}\pgfsys@curveto{13.44528pt}{1.4143pt}{12.81207pt}{0.7811pt}{12.81207pt}{0.0pt}\pgfsys@curveto{12.81207pt}{-0.7811pt}{13.44528pt}{-1.4143pt}{14.22638pt}{-1.4143pt}\pgfsys@curveto{15.00748pt}{-1.4143pt}{15.64069pt}{-0.7811pt}{15.64069pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.22638pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{27.0216pt}{0.0pt}\pgfsys@curveto{27.0216pt}{0.7811pt}{26.3884pt}{1.4143pt}{25.6073pt}{1.4143pt}\pgfsys@curveto{24.8262pt}{1.4143pt}{24.193pt}{0.7811pt}{24.193pt}{0.0pt}\pgfsys@curveto{24.193pt}{-0.7811pt}{24.8262pt}{-1.4143pt}{25.6073pt}{-1.4143pt}\pgfsys@curveto{26.3884pt}{-1.4143pt}{27.0216pt}{-0.7811pt}{27.0216pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.6073pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;+\;\;\ldots$
(3.35)

The first and third diagrams are computed as in section 3.2, whereas the
second is a chain diagram, for which the integral is well known (see e.g.
[54]):

$\displaystyle\int\frac{d^{d}x_{3}}{(x_{13}^{2})^{\Delta_{1}}(x_{23}^{2})^{\Delta_{2}}}=\frac{\pi^{\frac{d}{2}}}{(x_{12}^{2})^{\Delta_{1}+\Delta_{2}-\frac{d}{2}}}\frac{\Gamma\\!\left(\frac{d}{2}-\Delta_{1}\right)}{\Gamma(\Delta_{1})}\frac{\Gamma\\!\left(\frac{d}{2}-\Delta_{2}\right)}{\Gamma(\Delta_{2})}\frac{\Gamma\\!\left(\Delta_{1}+\Delta_{2}-\frac{d}{2}\right)}{\Gamma(d-\Delta_{1}-\Delta_{2})}\,.$
(3.36)

Once again, we are interested in reducing $\phi_{a}\phi_{b}$ into irreducible
components. On one hand, we can form a vector and a symmetric-traceless
operator as follows:

$\displaystyle Z_{V}V_{\hat{a}}=\phi_{1}\phi_{\hat{a}}\,,\qquad
Z_{T}T_{\hat{a}\hat{b}}=\phi_{\hat{a}}\phi_{\hat{b}}-\frac{\delta_{\hat{a}\hat{b}}}{N-1}\phi_{\hat{c}}^{2}\,,$
(3.37)

with the following renormalization factors:

$\displaystyle
Z_{V}=1-\frac{\lambda}{(4\pi)^{2}}\frac{h^{2}+2}{3\varepsilon}+O(\lambda^{2})\,,\qquad
Z_{T}=1-\frac{\lambda}{(4\pi)^{2}}\frac{h^{2}+4}{6\varepsilon}+O(\lambda^{2})\,.$
(3.38)

Again, it is straightforward to extract the anomalous dimensions at the
critical point and the normalization of the two-point functions:

$\displaystyle\langle\,V_{\hat{a}}(\tau)\,V_{\hat{b}}(0)\,\rangle$
$\displaystyle=\delta_{\hat{a}\hat{b}}\,\frac{{\mathcal{N}}_{V}^{2}}{\tau^{2\Delta_{V}}}\,,\quad\quad{\mathcal{N}}_{V}^{2}=\kappa^{2}\left(1-\frac{2N+18}{N+8}\,\aleph\,\varepsilon+O(\varepsilon^{2})\right)\,,$
$\displaystyle\langle\,T_{\hat{a}\hat{b}}(\tau)\,T_{\hat{c}\hat{d}}(0)\,\rangle$
$\displaystyle=\mathbf{T}_{\hat{a}\hat{b},\hat{c}\hat{d}}\,\frac{{\mathcal{N}}_{T}^{2}}{\tau^{2\Delta_{T}}}\,,\quad{\mathcal{N}}_{T}^{2}=2\kappa^{2}\left(1-\frac{N+10}{N+8}\,\aleph\,\varepsilon+O(\varepsilon^{2})\right)\,.$
(3.39)

Here we have introduced a symmetric-traceless tensor which will be useful
later:

$\displaystyle\mathbf{T}_{\hat{a}\hat{b},\hat{c}\hat{d}}=\frac{1}{2}\delta_{\hat{a}\hat{c}}\delta_{\hat{b}\hat{d}}+\frac{1}{2}\delta_{\hat{a}\hat{d}}\delta_{\hat{b}\hat{c}}-\frac{\delta_{\hat{a}\hat{b}}\delta_{\hat{c}\hat{d}}}{N-1}\,.$
(3.40)

On the other hand, we can form two independent scalars $\phi_{1}^{2}$ and
$\phi_{a}^{2}$, which mix at $O(\varepsilon)$ due to quantum corrections. The
renormalized fields $s_{\pm}$ are defined as

$\displaystyle\begin{pmatrix}\phi_{1}^{2}\\\
\phi_{a}^{2}\end{pmatrix}=Z_{s}\begin{pmatrix}s_{-}\\\ s_{+}\end{pmatrix}\,,$
(3.41)

where the renormalization factor $Z_{s}$ is a two-by-two matrix. To determine
$Z_{s}$ one requires that three-point functions of renormalized fields

$\displaystyle\langle\,[\phi_{1}]\,[\phi_{1}]\,s_{\pm}\,\rangle\,,\quad\langle\,[\phi_{1}]\,t_{\hat{a}}\,s_{\pm}\,\rangle\,,\quad\langle\,t_{\hat{a}}\,t_{\hat{b}}\,s_{\pm}\,\rangle\,,\quad$
(3.42)

have no poles in $\varepsilon$. We explain how to compute these three-point
functions in section 3.2. Furthermore, one should require that $Z_{s}$ is such
that $s_{\pm}$ have anomalous dimensions that do not mix. Demanding this we
find the anomalous-dimension matrix

$\displaystyle\gamma=Z^{-1}_{s}\frac{\partial Z_{s}}{\partial\log
M}\Big{|}_{\text{fixed
point}}=\left(\begin{array}[]{cc}\frac{5N+36-\sqrt{N^{2}+40N+320}}{2N+16}&0\\\
0&\frac{5N+36+\sqrt{N^{2}+40N+320}}{2N+16}\\\
\end{array}\right)\varepsilon+O(\varepsilon^{2})\,.$ (3.45)

Our choice of $Z_{s}$ guarantees that the two-point functions are orthogonal,
as one usually requires in CFT:

$\displaystyle\langle\,s_{\pm}(\tau)\,s_{\pm}(0)\,\rangle=\frac{{\mathcal{N}}_{s_{\pm}}^{2}}{\tau^{2\Delta_{s_{\pm}}}}\,,\qquad\langle\,s_{\pm}(\tau)\,s_{\mp}(0)\,\rangle=0\,.$
(3.46)

The formulas for $Z_{s}$ and ${\mathcal{N}}_{s_{\pm}}^{2}$ are somewhat
complicated and not particularly illuminating, so instead of writing them
explicitly we attach them in a notebook.

####  3.2.4 Three-point functions

Having determined the scaling dimension and normalization of the defect
operators of interest, we are ready to compute some of their three-point OPE
coefficients $\lambda_{{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}_{3}}$.
Here we shall focus on parity-even operators, that have three-point functions
of the form

$\displaystyle\langle{\mathcal{O}}_{1}(\tau_{1}){\mathcal{O}}_{2}(\tau_{2}){\mathcal{O}}_{3}(\tau_{3})\rangle=\frac{{\mathcal{N}}_{{\mathcal{O}}_{1}}{\mathcal{N}}_{{\mathcal{O}}_{2}}{\mathcal{N}}_{{\mathcal{O}}_{3}}\,\lambda_{{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}_{3}}}{|\tau_{12}|^{\Delta_{1}+\Delta_{2}-\Delta_{3}}|\tau_{13}|^{\Delta_{1}+\Delta_{3}-\Delta_{2}}|\tau_{23}|^{\Delta_{2}+\Delta_{3}-\Delta_{1}}}\,.$
(3.47)

By choosing different external operators there are many three-point functions
that can be computed. Here we focus on a subset of them, which can be compared
to our numerical study or in future works.

#### The correlator $\langle\,\phi\,\phi\,\phi\,\rangle$

The simplest three-point functions are the ones that involve only the
fundamental scalar. These are zero at tree level, but receive a contribution
at order $O(\varepsilon)$ from the following diagram:

$\displaystyle\langle\,\phi_{a}(\tau_{1})\,\phi_{b}(\tau_{2})\,\phi_{c}(\tau_{3})\,\rangle\;\;=\;\;\leavevmode\hbox
to40.63pt{\vbox to12.91pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-1.6143pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{11.09575pt}{34.14322pt}{11.09575pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{19.91684pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@lineto{19.91684pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33115pt}{7.96674pt}\pgfsys@curveto{21.33115pt}{8.74783pt}{20.69794pt}{9.38104pt}{19.91684pt}{9.38104pt}\pgfsys@curveto{19.13574pt}{9.38104pt}{18.50253pt}{8.74783pt}{18.50253pt}{7.96674pt}\pgfsys@curveto{18.50253pt}{7.18564pt}{19.13574pt}{6.55243pt}{19.91684pt}{6.55243pt}\pgfsys@curveto{20.69794pt}{6.55243pt}{21.33115pt}{7.18564pt}{21.33115pt}{7.96674pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{7.96674pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.91684pt}{7.96674pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{27.0216pt}{0.0pt}\pgfsys@curveto{27.0216pt}{0.7811pt}{26.3884pt}{1.4143pt}{25.6073pt}{1.4143pt}\pgfsys@curveto{24.8262pt}{1.4143pt}{24.193pt}{0.7811pt}{24.193pt}{0.0pt}\pgfsys@curveto{24.193pt}{-0.7811pt}{24.8262pt}{-1.4143pt}{25.6073pt}{-1.4143pt}\pgfsys@curveto{26.3884pt}{-1.4143pt}{27.0216pt}{-0.7811pt}{27.0216pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.6073pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;+\;\;\ldots$
(3.48)

Because this diagram is proportional to a factor
$\lambda_{*}\propto\varepsilon$, it suffices to evaluate the integral in
exactly $d=4$:

$\displaystyle\int\frac{d\tau_{4}\,d^{4}x_{5}}{x^{2}_{15}x^{2}_{25}x^{2}_{35}x^{2}_{45}}=\frac{2\pi^{4}}{\sqrt{\tau^{2}_{12}\tau^{2}_{13}\tau^{2}_{23}}}\,.$
(3.49)

To compute this integral, we exploit that it is invariant under the one-
dimensional conformal group, so we use a frame where $\tau_{1}=0$,
$\tau_{2}=1$ and $\tau_{3}=\infty$. We start integrating over $\tau_{4}$, then
the orthogonal directions $\vec{x}_{5}^{2}$, and finally over $\tau_{5}$, all
the steps being elementary. Splitting the diagram (3.48) into irreducible
components under $O(N-1)$, and keeping track of all normalization factors, we
find the two OPE coefficients

$\displaystyle\lambda_{\phi_{1}\phi_{1}\phi_{1}}=\frac{3\pi\varepsilon}{\sqrt{N+8}}+O(\varepsilon^{2})\,,\qquad\lambda_{tt\phi_{1}}=\frac{\pi\varepsilon}{\sqrt{N+8}}+O(\varepsilon^{2})\,.$
(3.50)

#### The correlator $\langle\,\nabla\phi\,\nabla\phi\,\phi\,\rangle$

We can also consider a three-point function where two of the external
operators are orthogonal derivatives
$\langle\nabla\phi_{a}(\tau_{1})\nabla\phi_{b}(\tau_{2})\phi_{c}(\tau_{3})\rangle$.
The same diagram as in (3.48) contributes, but now it leads to the integral

$\displaystyle\lim_{\vec{x}_{1},\vec{x}_{2}\to
0}\frac{\partial}{\partial\vec{x}_{1,i}}\frac{\partial}{\partial\vec{x}_{2,j}}\int\frac{d\tau_{4}\,d^{4}x_{5}}{x^{2}_{15}x^{2}_{25}x^{2}_{35}x^{2}_{45}}=\frac{4\pi^{4}/3}{\sqrt{\tau^{6}_{12}\tau^{2}_{13}\tau^{2}_{23}}}\,,$
(3.51)

which has been computed as before in eq. (3.49). Splitting the result into
irreducible components under $O(N-1)$ and keeping track of all normalization
factors, we find the three OPE coefficients

$\displaystyle\lambda_{DD\phi_{1}}=\frac{\pi\varepsilon}{\sqrt{N+8}}+O(\varepsilon^{2})\,,\quad\lambda_{\text{D}\nabla\phi
t}=\frac{\pi\varepsilon}{3\sqrt{N+8}}+O(\varepsilon^{2})\,,\quad\lambda_{\nabla\phi\nabla\phi\phi_{1}}=\frac{\pi\varepsilon}{3\sqrt{N+8}}+O(\varepsilon^{2})\,.$
(3.52)

#### The correlator $\langle\,\phi\,\phi\,\phi^{2}\,\rangle$

The last type of OPE coefficient we consider involves two fundamental fields
and a composite one:

$\displaystyle\langle\,\phi_{a}(\tau_{1})\,\phi_{b}(\tau_{2})\,\phi_{c}\phi_{d}(\tau_{3})\,\rangle\;\;=\;\;\leavevmode\hbox
to52.01pt{\vbox to16.13pt{\pgfpicture\makeatletter\hbox{\hskip
11.78091pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-5.69046pt}{0.0pt}\pgfsys@curveto{-5.69046pt}{15.5339pt}{34.14322pt}{15.5339pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{11.09575pt}{34.14322pt}{11.09575pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;+\;\;\leavevmode\hbox
to52.01pt{\vbox to16.13pt{\pgfpicture\makeatletter\hbox{\hskip
11.78091pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
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}{}\pgfsys@moveto{-5.69046pt}{0.0pt}\pgfsys@curveto{-5.69046pt}{15.5339pt}{34.14322pt}{15.5339pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{14.22638pt}{11.38092pt}\pgfsys@lineto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
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}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.6073pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;+\;\;\ldots$
(3.53)

The first and third diagrams are elementary, and have been computed in section
3.2. The second diagram contains a new integral that reads

$\displaystyle\int\frac{d^{d}x_{4}}{(x^{2}_{14}x^{2}_{24})^{1-\frac{\varepsilon}{2}}(x^{2}_{34})^{2-\varepsilon}}=\frac{\pi^{2}}{\tau^{2}_{13}\tau^{2}_{23}}\left(\frac{2}{\varepsilon}+3-\aleph+\log\left(\frac{\tau^{4}_{13}\tau^{4}_{23}}{\tau^{2}_{12}}\right)+O(\varepsilon)\right)\,.$
(3.54)

To compute it, we first rewrite the integral in parametric form using
Schwinger’s representation, and then we partial integrate as described in [55,
56].121212Although we have not used HyperInt [57], the package automatizes
this in the function dimregPartial. The result is expanded to order
$O(\varepsilon^{0})$, and each of the terms in the expansion is a convergent
integral that can be solved with elementary methods.

Once again, the correlator in eq. (3.53) contains several OPE coefficients. If
we let the third operator be a scalar under $O(N-1)$, then we have the
following OPE coefficients:

$\displaystyle\lambda_{\phi_{1}\phi_{1}s_{\pm}}$ $\displaystyle=\frac{\pm
2\sqrt{N-1}}{\sqrt{N^{2}+40N+320\mp(N+18)\sqrt{N^{2}+40N+320}}}\Bigg{(}1\pm\frac{(95N-640)\varepsilon}{64\sqrt{N^{2}+40N+320}}$
$\displaystyle\quad\pm\frac{17\varepsilon\sqrt{N^{2}+40N+320}}{64(N+8)}-\frac{7N^{3}+182N^{2}+1160N+2280}{4(N+8)\left(N^{2}+40N+320\right)}\varepsilon+O(\varepsilon^{2})\Bigg{)}\,,$
(3.55) $\displaystyle\lambda_{tts_{\pm}}$
$\displaystyle=\frac{\left(\mp(N+18)+\sqrt{N^{2}+40N+320}\right)(N-1)^{-1/2}}{\sqrt{N^{2}+40N+320\mp(N+18)\sqrt{N^{2}+40N+320}}}\Bigg{(}1\mp\frac{(95N-640)\varepsilon}{64\sqrt{N^{2}+40N+320}}$
$\displaystyle\quad\mp\frac{49\varepsilon\sqrt{N^{2}+40N+320}}{64(N+8)}-\frac{\left(11N^{3}+374N^{2}+3720N+12520\right)\varepsilon}{4(N+8)\left(N^{2}+40N+320\right)}+O(\varepsilon^{2})\Bigg{)}\,.$
(3.56)

If we instead let the third operator be a vector or a symmetric-traceless
tensor, we find:

$\displaystyle\lambda_{\phi_{1}tV}=1-\frac{\varepsilon}{N+8}+O(\varepsilon^{2})\,,\qquad\lambda_{ttT}=\sqrt{2}\left(1-\frac{\varepsilon}{N+8}+O(\varepsilon^{2})\right)\,.$
(3.57)

The normalization of $\lambda_{ttT}$ is chosen such that the tensor structure
in eq. (3.40) multiplies the three-point function in eq. (3.47).

####  3.2.5 Four-point functions

In this final section, we turn our attention to four-point functions of
operators formed by the fundamental field and possibly one transverse
derivative. Because of the OPE, four-point functions contain information about
three-point OPE coefficients, so we will be able to check some calculations of
section 3.2.4 and obtain new results.

#### The correlator $\langle\,\phi\,\phi\,\phi\,\phi\,\rangle$

The simplest four-point function is that of the fundamental field:

$\displaystyle\langle\,\phi_{a}(\tau_{1})\,\phi_{b}(\tau_{2})\,\phi_{c}(\tau_{3})\,\phi_{d}(\tau_{4})\,\rangle\;\;=\;\;\leavevmode\hbox
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{32.72049pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;+\;\;\ldots$
(3.58)

The first and third contributions lead to the disconnected part of the
correlator, while the non-trivial part is given by the second diagram. Since
the second diagram is multiplied by a coupling $\lambda_{*}=O(\varepsilon)$,
it suffices to evaluate the integral in $d=4$:

$\displaystyle\lim_{\vec{x}_{n}\to
0}\int\frac{d^{4}x_{5}}{x^{2}_{15}x^{2}_{25}x^{2}_{35}x^{2}_{45}}=-2\pi^{2}\frac{I_{1}(\xi)}{\tau^{2}_{12}\tau^{2}_{34}}\,.$
(3.59)

Since this integral preserves the one-dimensional conformal group, we can
evaluate it in the frame
$(\tau_{1},\tau_{2},\tau_{3},\tau_{4})=(0,\xi,1,\infty)$, where the cross-
ratio $\xi$ is defined in eq. (2.13). We integrate first over orthogonal
directions $\vec{x}_{5}^{2}$ and then over $\tau_{5}$, considering separately
the intervals $\tau_{5}\in(-\infty,0)$, $\tau_{5}\in(0,\xi)$,
$\tau_{5}\in(\xi,1)$ and $\tau_{5}\in(1,\infty)$. The final result reads

$\displaystyle I_{1}(\xi)=\xi\log(1-\xi)+\frac{\xi^{2}}{1-\xi}\log\xi\,.$
(3.60)

As usual, we decompose the correlator in eq. (3.58) into irreducible
components, and using the appropriate renormalization factors we obtain three
inequivalent correlators:

$\displaystyle G_{\phi_{1}\phi_{1}\phi_{1}\phi_{1}}(\xi)$
$\displaystyle=1+\xi^{2\Delta_{\phi_{1}}}+\left(\frac{\xi}{1-\xi}\right)^{2\Delta_{\phi_{1}}}+\frac{6\varepsilon
I_{1}(\xi)}{N+8}+O(\varepsilon^{2})\,,$ $\displaystyle
G_{\phi_{1}\phi_{1}t_{\hat{a}}t_{\hat{b}}}(\xi)$
$\displaystyle=\delta_{\hat{a}\hat{b}}+\varepsilon\,\frac{2\delta_{\hat{a}\hat{b}}}{N+8}I_{1}(\xi)+O(\varepsilon^{2})\,,$
$\displaystyle G_{t_{\hat{a}}t_{\hat{b}}t_{\hat{c}}t_{\hat{d}}}(\xi)$
$\displaystyle=\delta_{\hat{a}\hat{b}}\delta_{\hat{c}\hat{d}}+\delta_{\hat{a}\hat{c}}\delta_{\hat{b}\hat{d}}\xi^{2}+\delta_{\hat{a}\hat{d}}\delta_{\hat{b}\hat{c}}\left(\frac{\xi}{1-\xi}\right)^{2}+\frac{2\varepsilon
I_{1}(\xi)}{N+8}\Big{(}\delta_{\hat{a}\hat{b}}\delta_{\hat{c}\hat{d}}+\text{perms}\Big{)}+O(\varepsilon^{2})$
$\displaystyle=\delta_{\hat{a}\hat{b}}\delta_{\hat{c}\hat{d}}\,G_{tttt}^{S}(\xi)+(\delta_{\hat{a}\hat{d}}\delta_{\hat{b}\hat{c}}-\delta_{\hat{a}\hat{c}}\delta_{\hat{b}\hat{d}})\,G_{tttt}^{A}(\xi)+\mathbf{T}_{\hat{a}\hat{b},\hat{c}\hat{d}}\,G_{tttt}^{T}(\xi)\,.$
(3.61)

The correlator of four tilt operators decomposes into singlet $(S)$,
antisymmetric ($A$) and symmetric-traceless ($T$) channels with respect to the
$O(N-1)$ symmetry, where the last tensor structure is shown in eq. (3.40).

The virtue of having four-point functions is that they can be expanded in
conformal blocks to obtain anomalous dimensions and three-point coefficients.
For the vector, symmetric-traceless and antisymmetric channels we find:

$\displaystyle G_{t\phi_{1}\phi_{1}t}(\xi)$
$\displaystyle=\left(1-\frac{2\varepsilon}{N+8}\right)g^{\Delta_{t\phi_{1}},\Delta_{\phi_{1}t}}_{2}(\xi)+\varepsilon\,\frac{N+10}{N+8}\,\partial_{\Delta}g^{\Delta_{t\phi_{1}},\Delta_{\phi_{1}t}}_{2}(\xi)+\ldots\,,$
$\displaystyle G^{S}_{tttt}(\xi)$
$\displaystyle=\left(2-\frac{4\varepsilon}{N+8}\right)g_{2}(\xi)+\frac{4\varepsilon}{N+8}\,\partial_{\Delta}g_{2}(\xi)+\ldots\,,$
$\displaystyle G^{A}_{tttt}(\xi)$ $\displaystyle=g_{3}(\xi)+\ldots\,.$ (3.62)

The ellipses stand for higher dimensional operators, as well as higher order
corrections in $\varepsilon$. The first two lines confirm our computations of
$\Delta_{V}$, $\Delta_{T}$, $\lambda_{\phi_{1}tV}$ and $\lambda_{ttT}$. From
the third line, we conclude that the antisymmetric operator
$A_{ab}\sim\phi_{[a}\partial_{\tau}\phi_{b]}$ has scaling dimension
$\Delta_{A}=3+O(\varepsilon^{2})$ and three-point coefficient
$\lambda_{ttA}=1+O(\varepsilon^{2})$.

Of course, we can also expand correlators in the singlet channel, for example

$\displaystyle G_{\phi_{1}\phi_{1}\phi_{1}\phi_{1}}(\xi)$
$\displaystyle=1+\left(2-\frac{6\varepsilon}{N+8}\right)g_{2}(\xi)+\varepsilon\,\frac{4N+38}{N+8}\,\partial_{\Delta}g_{2}(\xi)+\ldots$
(3.63)

However, the above contributions are due to two operators $s_{\pm}$ with
nearly-degenerate dimension, so the expansion does not fix the scaling
dimension or OPE coefficient, but instead it relates them in a non-trivial
way:

$\displaystyle(\Delta_{s_{+}}-2)\lambda_{\phi_{1}\phi_{1}s_{+}}^{2}+(\Delta_{s_{-}}-2)\lambda_{\phi_{1}\phi_{1}s_{-}}^{2}=\varepsilon\,\frac{4N+38}{N+8}+O(\varepsilon^{2})~{},$
$\displaystyle\lambda_{\phi_{1}\phi_{1}s_{+}}^{2}+\lambda_{\phi_{1}\phi_{1}s_{-}}^{2}=2-\frac{6\varepsilon}{N+8}+O(\varepsilon^{2})\,.$
(3.64)

It is reassuring that eq. (3.15) and eq. (3.55) indeed satisfy these
relations. In a similar way, we can expand $G_{tttt}^{\text{sing}}$ and
$G_{\phi_{1}\phi_{1}tt}$, finding again perfect agreement with our results.
Note that the four-point functions do not capture the OPE coefficients in eq.
(3.50) at this order in $\varepsilon$.

#### The correlators $\langle\,\nabla\phi\,\nabla\phi\,\phi\,\phi\,\rangle$
and $\langle\,\nabla\phi\,\nabla\phi\,\nabla\phi\,\nabla\phi\,\rangle$

In a similar way, one can consider four-point functions that include
derivative operators $\nabla\phi_{a}$, and the only non-trivial contribution
is a contact diagram. For the case of two derivative operators, the relevant
integral is

$\displaystyle\lim_{\vec{x}_{n}\to
0}\frac{\partial^{2}}{\partial\vec{x}_{3,i}\partial\vec{x}_{4,j}}\int\frac{d^{4}x_{5}}{x^{2}_{15}x^{2}_{25}x^{2}_{35}x^{2}_{45}}=\frac{2\pi^{2}}{3}\frac{I_{2}(\xi)}{\tau_{12}^{2}\tau_{34}^{4}}\delta^{ij}\,,$
$\displaystyle
I_{2}(\xi)=\frac{\xi^{2}}{1-\xi}-(\xi+2)\xi\log(1-\xi)+\frac{\xi^{4}}{(1-\xi)^{2}}\log\xi\,,$
(3.65)

which has been computed using the same technique as in eq. (3.60). From this,
we can read off several correlators, for example

$\displaystyle G_{\phi_{1}\phi_{1}\text{D}_{i}\text{D}_{j}}(\xi)$
$\displaystyle=\delta_{ij}\left(1-\frac{\varepsilon
I_{2}(\xi)}{N+8}+O(\varepsilon^{2})\right)\,,$ $\displaystyle
G_{t_{\hat{a}}t_{\hat{b}}\text{D}_{i}\text{D}_{j}}(\xi)$
$\displaystyle=\delta_{\hat{a}\hat{b}}\delta_{ij}\left(1-\frac{\varepsilon
I_{2}(\xi)}{3(N+8)}+O(\varepsilon^{2})\right)\,.$ (3.66)

We can also obtain correlators with $\nabla\phi_{\hat{a}}$, but we do not
write them for compactness. Finally, if we consider four derivative operators
the relevant integral is

$\displaystyle\lim_{\vec{x}_{n}\to