DESY-24-099

Conformal line defects at finite temperature

Julien<EMAIL_ADDRESS>Bartomeu<EMAIL_ADDRESS>Enrico<EMAIL_ADDRESS>
Alessio<EMAIL_ADDRESS>and Elli
<EMAIL_ADDRESS>
a Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
b Departament de Física Quàntica i Astrofísica i Institut de Ciències del
Cosmos,
Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Catalonia,
Spain
c Mathematical Institute, University of Oxford,
Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, U.K.

Abstract

###### Abstract

We study conformal field theories at finite temperature in the presence of a
temporal conformal line defect, wrapping the thermal circle, akin to a
Polyakov loop in gauge theories. Although several symmetries of the conformal
group are broken, the model can still be highly constrained from its features
at zero-temperature. In this work we show that the defect and bulk one and
two-point correlators can be written as functions of zero-temperature data and
thermal one-point functions (defect and bulk). The defect one-point functions
are new data and they are induced by thermal effects of the bulk. For this new
set of data we derive novel sum rules and establish a bootstrap problem for
the thermal defect one-point functions from the KMS condition. We also comment
on the behaviour of operators with large scaling dimensions. Additionally, we
relate the free energy and entropy density to the OPE data through the one-
point function of the stress-energy tensor. Our formalism is validated through
analytical computations in generalized free scalar field theory, and we
present new predictions for the $\mathrm{O}(N)$ model with a magnetic impurity
in the $\varepsilon$-expansion and the large $N$ limit.

###### Contents

  1. 1 Introduction
  2. 2 Defect CFT at finite temperature
    1. 2.1 Line defects on the thermal manifold
    2. 2.2 Correlation functions
  3. 3 Setting up a bootstrap problem
    1. 3.1 From the KMS condition to sum rules
    2. 3.2 Heavy operators
  4. 4 Defect thermodynamics
    1. 4.1 Free energy density and entropy
    2. 4.2 The free energy of a quark
  5. 5 Applications
    1. 5.1 Magnetic lines in generalized free field theory
    2. 5.2 The magnetic line in the $\mathrm{O}(N)$ model
  6. 6 Conclusions and outlook
  7. A Broken Ward identities
  8. B A comment on the real time formalism
  9. C Defect thermodynamics in $d=2$
  10. D Stress-energy tensor and relevant operators on the defect

##  1 Introduction

Conformal Field Theory (CFT) finds applications across various areas of
physics, ranging from condensed-matter systems at the critical point to
quantum gravity via the AdS/CFT correspondence. The study of defects within
CFTs is also fundamental since these describe a wide range of physical
phenomena, such as magnetic impurities in materials with emergent Lorentz
symmetry (see e.g. Dias:2013bwa ; Allais:2014fqa  and references therein), or
the radiation of moving quarks in high-energy physics Correa:2012at ;
Fiol:2012sg ; Lewkowycz:2013laa . A notable class of defects, called conformal
defects, breaks the conformal symmetry in a controlled manner, with the
residual symmetry imposing significant constraints on observables. This has
led to the development of the defect bootstrap program Liendo:2012hy ;
Billo:2016cpy ; Lemos:2017vnx ; Gaiotto:2013nva ; Gliozzi:2015qsa . However,
much less is known about these models at finite temperature. In this case, a
scale (the inverse temperature $\beta$) is introduced by considering the
defect CFT on the geometry $S_{\beta}^{1}\times\mathbb{R}^{d-1}$. This paper
focuses on the scenario of a temporal line defect, wrapping the thermal circle
$S^{1}_{\beta}$, inspired by its relevance to the confinement problem and the
AdS/CFT correspondence. Indeed, in gauge theories, the thermal analogue of the
Wilson loop, known as the Polyakov loop, acts as an order parameter for the
confinement/deconfinement phase transition Polyakov:1978vu . Although
confinement does not occur in a CFT, studying this setup provides a context
endowed with analytical tools. Additionally, thermal CFTs are holographically
dual to black holes in AdS, making the study of defects in this framework both
fascinating and crucial Witten:1998zw . In condensed-matter physics, line
defects wrapping the thermal circle describe point-like impurities, which can
be experimentally probed Affleck:1995ge ; Sachdev:2010uj .

Although the conformality of the bulk theory is broken by both thermal effects
and the presence of the defect, many local properties of the zero temperature
CFT remain useful for characterizing the system. Thermal correlation functions
(without a defect) can be analyzed using the Operator Product Expansion (OPE),
albeit with a finite radius of convergence and additional OPE data in the form
of thermal one-point functions. At zero temperature, the associativity of the
OPE leads to highly constraining consistency conditions. At finite
temperature, an equivalent condition was identified as the Kubo-Martin-
Schwinger (KMS) condition El-Showk:2011yvt ; Iliesiu:2018fao . This approach
is central to the bootstrap methods for thermal CFTs, initiated in the seminal
work Iliesiu:2018fao  and further developed in subsequent studies
David:2023uya ; David:2024naf ; Iliesiu:2018zlz ; Marchetto:2023fcw ;
Marchetto:2023xap . Similarly, the bootstrap approach to defect CFTs at zero
temperature introduces new defect OPE data. Specifically, the presence of a
conformal line defect results in a one-dimensional CFT living on the defect.
Such setups have received significant attention recently, with bootstrap
methods yielding a wealth of novel results Liendo:2018ukf ; deLeeuw:2017dkd ;
Liendo:2019jpu ; Bianchi:2020hsz ; Gimenez-Grau:2022czc ; Ferrero:2021bsb ;
Barrat:2021yvp ; Cavaglia:2021bnz ; Drukker:2022pxk ; Gimenez-Grau:2022ebb ;
Pannell:2023pwz .

This work aims to construct a bootstrap framework for a thermal defect CFT,
specifically focusing on a line defect wrapping the thermal circle. We find
that the minimal set of OPE data required to fully characterize a specific
model is:

  1. 1.

Zero temperature CFT data: the scaling dimensions $\Delta_{\mathcal{O}}$ and
the structure constants $\mu_{ijk}$ of the bulk CFT;

  2. 2.

Finite temperature data: the thermal one-point functions of bulk operators
$b_{\mathcal{O}}$ (in the absence of the defect);

  3. 3.

Zero temperature defect data: the scaling dimensions
$\widehat{\Delta}_{\widehat{\mathcal{O}}}$ and the structure constants
$\widehat{\mu}_{ijk}$ of the defect CFT, as well as the bulk-to-defect
coefficients $\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}$;

  4. 4.

Thermal+defect data: the new data consists of the one-point functions of
defect operators $\widehat{b}_{\widehat{\mathcal{O}}}$.

The CFT and defect data are inherited from the zero temperature theory and
remain unaltered by thermal effects. Thermal one-point functions of bulk
operators are known only for a few specific cases, but they are the subject of
the currently developing finite temperature bootstrap program. In our
approach, these three classes of data will be considered as input. In this
work, the one-point functions of the one-dimensional theory on the defect are
the new data we are interested to study. As we shall explain, these defect
one-point functions are a consequence of the thermal excitations of the bulk.
Therefore the $1d$ theory living on the defect is more than just a thermal
CFT.

The paper is structured as follows:

  * $\star$

in Section 2, we start with an analysis of the symmetries of the system, and
in particular we show how correlation functions are constrained by the OPE
data listed above. We provide an explicit expansion for one-point functions of
bulk operators and present an inversion formula to extract the defect one-
point functions;

  * $\star$

in Section 3, we set up a bootstrap problem for the new data
$\widehat{b}_{\widehat{\mathcal{O}}}$, by deriving explicit sum rules from the
KMS condition imposed on a two-point function of identical bulk operators in
the presence of the defect. We discuss some implications and how the behaviour
of heavy operators can be predicted;

  * $\star$

in Section 4, we study the thermodynamics of the system in $d>2$. In
particular, we relate the free energy and the entropy density of the system to
the defect OPE of the stress-energy tensor. We also comment on the defect
entropy and the free energy of a moving quark in gauge theories;

  * $\star$

in Section 5, we present applications to specific models. We test our
equations in the case of Generalized Free (scalar) Field (GFF) theory, and we
obtain new results for the case of the $d$-dimensional $\mathrm{O}(N)$ model,
both in the contexts of the $\varepsilon$-expansion and at large $N$.

Finally, in Section 6, we conclude with a discussion of our results and a
presentation of potential future prospects. In Appendix A, we present the
(broken) Ward identities for the conformal group in an explicit form, while in
Appendix B, we comment on the real-time formalism, which could provide a
different approach to the same problem. In Appendix C we review the defect
thermodynamics of two-dimensional CFTs and finally in Appendix D we present a
technical aspect regarding the bulk to defect OPE of the stress-energy tensor
which is important for the finiteness of the entropy in the zero temperature
limit.

##  2 Defect CFT at finite temperature

In this Section, we discuss the symmetries of a CFT with a defect at finite
temperature. We present possible configurations and analyze which symmetries
are broken or preserved when the defect wraps the thermal circle. We then
examine the correlation functions associated with the thermal defect CFT and
identify the OPE data required for a complete characterization. We conclude by
presenting an inversion formula for the defect one-point functions.

###  2.1 Line defects on the thermal manifold

####  2.1.1 Configurations

Our starting point is a Euclidean CFT at zero temperature, i.e. a CFT on the
flat manifold $\mathbb{R}^{d}$. We label the representations of the conformal
group by the scaling dimensions $\Delta$ and the spin $J$. A defect CFT can be
defined by inserting an extended operator of codimension $q$ (called a
conformal defect) that preserves part of the original symmetry. Specifically,
for a conformal line defect, the codimension is $q=d-1$ and the symmetry is
broken in the following way

$\mathrm{SO}(d+1,1)\longrightarrow\mathrm{SO}(2,1)\times\mathrm{SO}(d-1)\,.$
(1)

The $\mathrm{SO}(2,1)$ group can be interpreted as the one-dimensional
conformal symmetry preserved along the defect, while $\mathrm{SO}(d-1)$
corresponds to the unbroken rotations around the defect. From the $1d$
perspective, the latter can be viewed as a global symmetry Kapustin:2005py ;
Drukker:2005af . Defect representations are then labeled by the quantum
numbers ${\widehat{\Delta}}$, the $1d$ scaling dimension, and $s$, the
transverse spin, associated to the group $\mathrm{SO}(d-1)$.

Our goal is to study this system at finite temperature. More precisely, we
consider the defect CFT not on flat space, but on the thermal manifold

${\mathcal{M}}_{\beta}=S^{1}_{\beta}\times{\mathbb{R}}^{d-1}\ ,$ (2)

where the original time direction is compactified along a circle of length
$\beta=1/T$, with $T$ being the temperature. In the absence of a defect, this
compactification explicitly breaks the conformal symmetry of the bulk theory
Iliesiu:2018fao ; Marchetto:2023fcw . Contrary to the zero temperature case,
the orientation of the line defect plays a significant role in how the
symmetries are broken or preserved. We can consider three different systems:

  1. (a)

The line defect wraps the thermal circle;

  2. (b)

The line defect is placed along one of the (not compactified) space
directions;

  3. (c)

The line defect is placed in a direction that includes both a component in
time and space. In two dimensions, this defect can for instance take the shape
of a spiral wrapping the thermal cylinder.

Graphical representations of cases (a) and (b) are shown in Fig. 1. At zero
temperature, all configurations are equivalent in Euclidean space. However, at
finite temperature, these different setups describe distinct physical systems.
For instance, in setup (a), the defect preserves translations along the time
direction and preserves a symmetry group $\mathrm{SO}(d-1)$. In contrast, in
setup (b), only rotations orthogonal to the defect remain unbroken, reducing
the symmetry group to $\mathrm{SO}(d-2)$.

$\vec{x}$$\tau$${\mathcal{P}}$$\vec{x}=0$

$\vec{x}$$\tau$ ${\mathcal{W}}$$\tau=0$

Figure 1:  _The left figure shows a temporal line defect wrapping the thermal
circle at the spatial position $\vec{x}=0$. This configuration corresponds to
case_ (a) _in the main text, and is the focus of this paper. On the right
side, the setup_ (b) _is illustrated, for which the spatial line defect is
placed at $\tau=0$ and extends in a spatial direction._

Although all configurations have significant physical applications, this paper
focuses on type (a) which maximizes the residual conformal symmetry of the
defect CFT. Importantly, in the context of gauge theories, this configuration
corresponds to Polyakov loops, whose one-point function acts as the order
parameter for the confinement/deconfinement transition.

####  2.1.2 Broken and unbroken symmetries

Let us summarize the effect of turning on the temperature on the symmetries of
the conformal group for the case where the defect wraps the thermal circle.
Building upon the discussion of the previous Section, we have:

  * $\star$

Translations: Time translations around the thermal circle are unbroken, while
spatial translations are broken by the presence of the defect.

  * $\star$

Rotations: Spatial rotations are unbroken, while boosts are broken by both the
defect and finite temperature effects.

  * $\star$

Dilatations and SCTs: Dilatations and special conformal transformations (SCTs)
are broken only by finite temperature effects.

  * $\star$

Global symmetries: Any global symmetry is preserved at finite temperature, but
can be broken by the defect. The potential breaking of external symmetries by
the defect depends on the details of the theory.

This set of symmetries defines the thermal defect CFT that we study in the
rest of this work. It is possible to derive (broken) Ward identities in the
spirit of Marchetto:2023fcw . This results in non-trivial constraints on the
correlation functions of the theory. These (broken) Ward identities are
derived and presented in Appendix A.

###  2.2 Correlation functions

We now turn our attention to the implications of these symmetries: our aim is
to provide the OPE data necessary to (in principle) fully solve the system
through repeated use of the bulk and defect OPEs.

####  2.2.1 Bulk and defect OPEs

One crucial aspect of this setup is that the thermal geometry described by (2)
is conformally flat.666A manifold is conformally flat if for each point there
is a neighborhood that can be map to an open subset of the flat space via a
conformal transformation. This does not imply the existence of a conformal map
between the full thermal manifold and the flat space: this only happens in
$d=2$. This implies that local properties of the zero temperature theory are
preserved when the time dimension is compactified. In other words, the scaling
dimensions of the operators remain unaffected, and products of two (or more)
operators can be expanded using the same OPE as at zero temperature. For
instance, the _bulk OPE_ for two identical scalar operators $\phi$ of
dimension $\Delta_{\phi}$ is given by

$\phi(x_{1})\phi(x_{2})=\frac{1}{x_{12}^{2\Delta_{\phi}}}\sum_{{\mathcal{O}}\in\phi\times\phi}\mu_{\phi\phi{\mathcal{O}}}|x_{12}|^{\Delta-J}x_{12\,\mu_{1}}\ldots
x_{12\,\mu_{J}}{\mathcal{O}}^{\mu_{1}\ldots\mu_{J}}(x_{2})\,,$ (3)

where the conformal data (i.e., the scaling dimensions $\Delta$ and the
structure constants $\mu_{\phi\phi{\mathcal{O}}}$) appearing in this equation
are zero temperature data. Before proceeding let us notice that the OPE
convergence, discussed in similar CFT contexts in Pappadopulo:2012jk ;
Antunes:2021qpy ; Cuomo:2024vfk , is now limited to a finite region.
Concretely, the convergent region of the bulk OPE (3) in the presence of the
defect is given by

$\tau_{12}^{2}+\vec{x}_{12}^{2}\leq\min\left\\{\beta^{2},|\vec{x}_{1}|^{2},|\vec{x}_{2}|^{2}\right\\}\,,$
(4)

with $|\vec{x}_{i}|$ the transverse position of the two operators with respect
to the defect.

In defect CFT, it is well-known that bulk operators can be expanded in terms
of the defect operators introduced in Section 2.1.1, by bringing the bulk
operator close to the defect. For a scalar operator $\phi$, this results in
the _defect OPE_

$\phi(\tau,\vec{x})=\frac{1}{|\vec{x}|^{\Delta}}\sum_{{\widehat{\mathcal{O}}}}\lambda_{\phi{\widehat{\mathcal{O}}}}|\vec{x}|^{{\widehat{\Delta}}-s}x_{i_{1}}\ldots
x_{i_{s}}{\widehat{\mathcal{O}}}^{i_{1}\ldots i_{s}}(\tau)\,,$ (5)

where $|\vec{x}|$ is the transverse distance between the operator and the
defect. The OPE written above is expected to be convergent in the region
$|\vec{x}|<\beta$, unless other operators are closer to the defect. Once
again, the OPE data appearing in this equation corresponds to the zero
temperature case. Here, the coefficients
$\lambda_{\phi{\widehat{\mathcal{O}}}}$ describe bulk-defect two-point
functions, which are fixed kinematically at $T=0$ Billo:2016cpy .

####  2.2.2 Bulk and defect one-point functions

Non-local properties of the theory can however be affected by the
compactification of the time dimension. Since dilatations are not preserved,
one-point functions of local operators (without defects) can in general be
non-zero. Translational invariance along the thermal circle fixes them to be
constant and depend on a single coefficient Iliesiu:2018fao ;
Marchetto:2023fcw

$\langle\,{\mathcal{O}}_{\Delta}^{\mu_{1}\ldots\mu_{J}}\,\rangle_{\beta}=\frac{b_{\mathcal{O}}}{\beta^{\Delta}}\left(e^{\mu_{1}}\ldots
e^{\mu_{J}}-\text{traces}\right)\,,$ (6)

where $e^{\mu}$ is non-zero only when placed in the time direction. We refer
to these correlators as _bulk_ thermal one-point functions.

In order to understand the OPE data needed to characterize the thermal CFT in
the presence of the defect, we study the first kinematically non-trivial
correlator, i.e. the one-point function of a bulk operator in presence of the
defect, that we denote by ${\mathcal{P}}$. At zero temperature, it is
kinematically fixed, but since translations orthogonal to the defect are
broken, the correlator is now given by

$\langle\,\phi(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\frac{F_{\phi}(z)}{|\vec{x}|^{\Delta_{\phi}}}\,,$
(7)

for a scalar operator $\phi$, with $z$ a dimensionless variable defined as

$z=\frac{|\vec{x}|}{\beta}\,.$ (8)

Using the defect OPE (5), we can rewrite (7) as

$\langle\,\phi(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\frac{1}{|\vec{x}|^{\Delta_{\phi}}}\sum_{{\widehat{\mathcal{O}}}}\lambda_{\phi{\widehat{\mathcal{O}}}}|\vec{x}|^{{\widehat{\Delta}}-s}x_{i_{1}}\ldots
x_{i_{s}}\langle{\widehat{\mathcal{O}}}^{i_{1}\ldots
i_{s}}(\tau){\mathcal{P}}\rangle_{\beta}\,,$ (9)

which is convergent for $|\vec{x}|\leq\beta$. As mentioned in Section 2.1.1,
the defect operators ${\widehat{\mathcal{O}}}$ are described by their quantum
numbers ${\widehat{\Delta}}$ and $s$, corresponding respectively to the
symmetry groups $\mathrm{SO}(2,1)$ and $\mathrm{SO}(d-1)$. Note that, at zero
temperature, the correlator on the right-hand side vanishes for all operators
except for the identity $\widehat{\mathds{1}}$. From the invariance under time
translations, we find that the defect one-point functions are given by

$\langle\,{\widehat{\mathcal{O}}}_{{\widehat{\Delta}}}^{i_{1}\ldots
i_{s}}\,\rangle_{\beta}=\frac{\widehat{b}_{{\widehat{\mathcal{O}}}}}{\beta^{{\widehat{\Delta}}}}\Pi^{i_{1}\ldots
i_{s}}\,,$ (10)

with $\Pi^{i_{1}\ldots i_{s}}$ a $\mathrm{SO}(d-1)$-invariant tensor
structure,777There is a normalization to choose in the definition of the
tensor structure. In order to simplify some of the following expression we
choose $x_{i_{1}}\ldots x_{i_{s}}\pi^{i_{1}\ldots i_{s}}=|\vec{x}|^{s}$.
similarly to the bulk case. We refer to these correlators as defect one-point
functions $\widehat{b}_{\widehat{\mathcal{O}}}$.

It should be emphasized at this point that these correlators do not correspond
to one-dimensional thermal one-point functions. To appreciate this fact let us
consider a $1d$ CFT compactified on $S_{\beta}^{1}$. In that case, the
existence of a conformal map between the infinite line $\mathbb{R}$ and the
circle $S^{1}_{\beta}$ and the absence of conformal anomalies immediately
imply that the thermal one-point functions vanish. Naively, this seems to be
in contradiction with (10) being non-trivial. However, thermal effects in the
bulk induce additional one-point functions to the one dimensional defect
theory living on $S_{\beta}^{1}$, as we will now explain.

As an intermediate step, we contract the $\mathrm{SO}(d-1)$ tensor structure
$\Pi^{i_{1}\ldots i_{s}}$ with $x_{i_{1}}\ldots x_{i_{s}}$

$x_{i_{1}}\ldots
x_{i_{s}}\langle\,{\widehat{\mathcal{O}}}_{{\widehat{\Delta}}}^{i_{1}\ldots
i_{s}}\,\rangle_{\beta}=\widehat{b}_{{\widehat{\mathcal{O}}}}\frac{|\vec{x}|^{s}}{\beta^{{\widehat{\Delta}}}}\,,$
(11)

then (7) can be expressed as

$F_{\phi}(z)=\sum_{{\widehat{\mathcal{O}}}}\lambda_{\phi{\widehat{\mathcal{O}}}}\,\widehat{b}_{{\widehat{\mathcal{O}}}}\,z^{{\widehat{\Delta}}}\,.$
(12)

As a side comment, notice that operators with the same conformal dimension but
different transverse spin appear with the same power of the dimensionless
ratio $z$. In interacting theories, degeneracies are effectively rare, and
considering several one-point functions of bulk operators can be used to lift
the degeneracy.

If now we consider the one-point function in the presence of a line defect(7),
the function $F_{\phi}(z)$ necessarily obeys two special limits

$F_{\phi}(z)\overset{z\to 0}{\sim}\lambda_{\phi\hat{\mathds{1}}}\,,\qquad
F_{\phi}(z)\overset{z\to\infty}{\sim}b_{\phi}z^{\Delta}\,.$ (13)

The first limit corresponds to the zero temperature case. Zero temperature
one-point functions of the defect operators are zero apart from the
identity.888We implicitly use the normalization
$\langle\,\widehat{\mathds{1}}\,\rangle=\langle\,\widehat{\mathds{1}}\,\rangle_{\beta}=1$.
The second limit instead corresponds to the high-temperature case. $b_{\phi}$
generically is non-zero: by looking at the equation (13), we deduce that the
defect one-point functions $\widehat{b}_{\widehat{\mathcal{O}}}$ can be non-
zero in general. Notice that the only physically meaningful quantity is the
ratio $z$, hence the two limits can also be seen respectively as short and
large distance from the line.

Keeping all the above in mind, any correlation function of this thermal defect
CFT can (in principle) be decomposed by using the bulk and defect OPEs, and be
expressed in terms of zero temperature data, namely the scaling dimensions and
structure constants, to which we need to add the thermal one-point functions
(without defect) and the defect one-point functions. For instance, two-point
functions of defect operators $\widehat{\phi}_{1}$ and $\widehat{\phi}_{2}$
can be expressed in terms of the coefficients
$\widehat{\mu}_{\widehat{\phi}_{1}\widehat{\phi}_{2}{\widehat{\mathcal{O}}}}$
and $\widehat{b}_{\widehat{\mathcal{O}}}$. This set of elementary OPE data is
summarized in Table 1.999At zero temperature, certain physical observables are
related to the normalization of the two-point functions of defect operators.
This is for instance the case of the Bremsstrahlung function, which is related
to the two-point function of the displacement operator. This normalization can
be thought of as a function of the bulk-to-defect OPE coefficients, and for
this reason, we choose not to insert them in the table. Let us comment that in
principle the zero temperature CFT data suffices to compute any thermal
correlation function since those are defined as correlators computed in the
state defined by the density matrix $\rho=e^{-\beta\bm{H}}$ (we review this
formalism in Appendix B). Nonetheless, using this fact explicitly is very
hard, even when the defect is not present Iliesiu:2018fao . On the contrary,
we show in Section 3 that a bootstrap problem for the new data
$\widehat{b}_{{\widehat{\mathcal{O}}}}$ is very similar to the case of the
thermal CFT without defect.

| bulk data | bulk-defect data | defect data
---|---|---|---
$T=0$ | $\Delta_{\mathcal{O}}$ | $\sim\langle\,{\mathcal{O}}{\mathcal{O}}\,\rangle$ | $\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}$ | $\sim\langle\,{\mathcal{O}}{\widehat{\mathcal{O}}}\,\rangle$ | ${\widehat{\Delta}}_{{\widehat{\mathcal{O}}}}$ | $\sim\langle\,{\widehat{\mathcal{O}}}{\widehat{\mathcal{O}}}\,\rangle$
$\mu_{ijk}$ | $\sim\langle\,{\mathcal{O}}_{i}{\mathcal{O}}_{j}{\mathcal{O}}_{k}\,\rangle$ |  |  | $\widehat{\mu}_{ijk}$ | $\sim\langle\,{\widehat{\mathcal{O}}}_{i}{\widehat{\mathcal{O}}}_{j}{\widehat{\mathcal{O}}}_{k}\,\rangle$
$T=1/\beta$ | $b_{{\mathcal{O}}}$ | $\sim\langle\,{\mathcal{O}}\,\rangle_{\beta}$ |  |  | $\widehat{b}_{{\widehat{\mathcal{O}}}}$ | $\sim\langle\,{\widehat{\mathcal{O}}}\,\rangle_{\beta}$
Table 1:  _Summary of the OPE data needed to characterize the thermal defect
CFT, and how the coefficients are related to correlation functions at zero and
finite temperature. Notice that the only two new sets of coefficients
appearing at $T\neq 0$ are the thermal bulk one-point functions
$b_{\mathcal{O}}$ and the defect one-point functions
$\widehat{b}_{\widehat{\mathcal{O}}}$._

####  2.2.3 An inversion formula for defect one-point functions

We have seen in the previous Section that defect one-point functions are the
fundamental quantities to know, but also that they are induced by the bulk
theory as thermal excitations. We now show how these coefficients can be
determined through a computation in the bulk. Indeed, it is possible to write
the OPE expansion of $F_{\phi}(z)$ (defined in (7)) as

$F_{\phi}(z)=\int_{-\varepsilon-i\infty}^{-\varepsilon+i\infty}\mathrm{d}\Delta\
a(\Delta)z^{\Delta}\,,$ (14)

where the function $a(\Delta)$ has poles at the scaling dimensions of the
physical operators, i.e. the residue is fixed to be

$a(\Delta)\sim-\frac{\lambda_{\phi{\widehat{\mathcal{O}}}}\,\widehat{b}_{{\widehat{\mathcal{O}}}}}{\Delta-{\widehat{\Delta}}}\,,$
(15)

where, as a technical requirement, $a(\Delta)$ is required not to grow
exponentially in the positive $\Delta$ half (complex) plane. Here,
$\varepsilon$ parametrizes the contour in the complex plane, which can be
chosen arbitrarily as far as the integral converges. It is possible to perform
a Laplace transform on (14) in order to obtain the function $a(\Delta)$

$a(\Delta)=\int_{0}^{1}\mathrm{d}z\ z^{\Delta-1}\ F_{\phi}(z)\,.$ (16)

From the expression (15), we obtain the inversion formula for the OPE
coefficients

$\lambda_{\phi{\widehat{\mathcal{O}}}}\,\widehat{b}_{{\widehat{\mathcal{O}}}}=\operatorname{Res}_{\Delta\to{\widehat{\Delta}}}\int_{0}^{1}\mathrm{d}z\
z^{\Delta-1}\ F_{\phi}(z)\,.$ (17)

This equation provides a way to extract defect one-point functions given a
scalar bulk one-point function and the zero temperature structure constants
$\lambda_{\phi{\widehat{\mathcal{O}}}}$.

To conclude, note that the operators are identified by their scaling
dimensions in (17). As mentioned in the previous Section, degeneracy in the
transverse spin cannot be resolved using bulk one-point functions. It is
however possible that two degenerate operators appear in two different bulk
computations, i.e., with different zero temperature coefficients, and in this
case the degeneracy can in principle be resolved.

##  3 Setting up a bootstrap problem

This Section is dedicated to formulating a bootstrap problem for the defect
one-point functions at finite temperature. We present novel sum rules derived
from the KMS condition, that impose strong constraints on the coefficients
$\widehat{b}_{\widehat{\mathcal{O}}}$. With this in mind, we will also discuss
the behaviour of heavy operators.

###  3.1 From the KMS condition to sum rules

$|\vec{x}|$$\tau$${\mathcal{P}}$$\phi$$\phi$

21

Figure 2:  _The configuration for the KMS condition in the collinear limit is
depicted on the left. Here, the two local operators are located at a distance
$|\vec{x}|$ from the Polyakov loop, and separated in the time direction by
$\tau$. The figure on the right shows the order in which the OPEs must be
taken in order to obtain the sum rules: (1) the bulk OPE
$\phi\times\phi\to\sum{\mathcal{O}}$, (2) the defect OPE
${\mathcal{O}}\times{\mathcal{P}}\to\sum{\widehat{\mathcal{O}}}$. The dotted
circle is here to emphasize the condition $\tau<|\vec{x}|<\beta$, necessary to
be in the appropriate OPE regime._

Our starting point is the KMS condition for two identical scalar operators
$\phi(\tau,\vec{x})$. In order to write an equation in which both sides can be
expanded in OPEs we can additionally use $\tau\to-\tau$.101010For parity
invariant theories $\tau\to-\tau$ is a symmetry. In general we can compose
with parity action on one of the spatial direction to preserve the same
$\mathbb{Z}_{2}$ symmetry Iliesiu:2018fao . Concretely,

$0=\langle\,\phi(\beta/2+\tau,\vec{x}_{1})\phi(0,\vec{x}_{2}){\mathcal{P}}\,\rangle_{\beta}-\langle\,\phi(\beta/2-\tau,\vec{x}_{1})\phi(0,\vec{x}_{2}){\mathcal{P}}\,\rangle_{\beta}\,.$
(18)

We describe in the following how to use the bulk and defect OPEs in order to
derive sum rules for the one-point function coefficients
$\widehat{b}_{\widehat{\mathcal{O}}}$.

The first step is to use the bulk OPE given in (3) on the two terms in (18).
Note that, for this expansion to be convergent, we need to place the operators
such that $|x_{12}|<\text{min}(|\vec{x}_{1}|,|\vec{x}_{2}|)<\beta$. We obtain
the following equation

$\sum_{{\mathcal{O}}\in\phi\times\phi}\mu_{\phi\phi{\mathcal{O}}}\langle\,{\mathcal{O}}^{\mu_{1}\ldots\mu_{J}}(\vec{x}_{2}){\mathcal{P}}\,\rangle_{\beta}|v_{+}|^{\Delta-2\Delta_{\phi}-J}v_{+\mu_{1}}\ldots
v_{+\mu_{J}}=\\\
=\sum_{{\mathcal{O}}\in\phi\times\phi}\mu_{\phi\phi{\mathcal{O}}}\langle\,{\mathcal{O}}^{\mu_{1}\ldots\mu_{J}}(\vec{x}_{2}){\mathcal{P}}\,\rangle_{\beta}|v_{-}|^{\Delta-2\Delta_{\phi}-J}v_{-\mu_{1}}\ldots
v_{-\mu_{J}}\,,$ (19)

where the vectors $v_{\pm}$ are defined through

$v_{\pm}^{\mu}:=(\beta/2\pm\tau,\vec{x}_{12})\,,$ (20)

which describe the distance between the two local operators. The expression
simplifies considerably if the local operators are placed on the same thermal
circle:111111In the conventions of, e.g. Lemos:2017vnx , this collinear limit
can be understood as setting the cross-ratios equal to one another, i.e.
$z={\bar{z}}$. Equivalently, it corresponds to $\cos\phi=0$ in the notation of
Billo:2016cpy .

$v_{\pm}^{\mu}=\delta^{0\mu}\biggl{(}\frac{\beta}{2}\pm\tau\biggr{)}\,.$ (21)

The corresponding configuration is represented in Fig. 2. The resulting
equation can then be expanded in $\tau$, and each term must vanish
individually. This yields

$0=\sum_{{\mathcal{O}}}\mu_{\phi\phi{\mathcal{O}}}\langle\,{\mathcal{O}}^{0\ldots
0}(\vec{x}){\mathcal{P}}\,\rangle_{\beta}\frac{\beta^{\Delta}}{2^{\Delta}}\binom{\Delta-2\Delta_{\phi}}{n}\,,$
(22)

with $n\in 2\mathbb{N}+1$, as the terms with even powers cancel exactly.

The next step is to perform the defect OPE given in (5) for the operators
${\mathcal{O}}^{0\ldots 0}(\vec{x})$. Note that the choice of placing the two
operators at the same distance from the defect, namely the choice in equation
(21), introduces two main simplifications: the operators effectively behave
like scalars in this limit, and (5) can be used (see also (7)) without
entering the problem of computing spinning blocks. Furthermore only primary
operators contribute in the equation (22) because of translational invariance
along the time direction. Using the notation defined in Table 1 for the OPE
coefficients, the following sum rules are obtained

$0=\sum_{{\mathcal{O}},{\widehat{\mathcal{O}}}}\mu_{\phi\phi{\mathcal{O}}}\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}\widehat{b}_{{\widehat{\mathcal{O}}}}\binom{\Delta-2\Delta_{\phi}}{n}\frac{z^{{\widehat{\Delta}}-\Delta}}{2^{\Delta}}\,,$
(23)

where $z$ is the cross-ratio defined in (8). This can be recast as

$0=\sum_{\Delta,\bar{\Delta}}\mu_{\phi\phi{\mathcal{O}}}\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}\widehat{b}_{{\widehat{\mathcal{O}}}}\binom{\Delta-2\Delta_{\phi}}{n}\frac{z^{\bar{\Delta}}}{2^{\Delta}}\,,\qquad\bar{\Delta}={\widehat{\Delta}}-\Delta\
.$ (24)

A summary of the order in which the OPEs have been applied is shown in Fig. 2.
These sum rules impose strong constraints on the coefficients
$\widehat{b}_{\widehat{\mathcal{O}}}$, which can be studied using a variety of
bootstrap techniques. For thermal bulk CFTs, analogous equations have been
presented in Marchetto:2023xap . Observe that in Marchetto:2023xap  a
generalization of these sum rules, in the case where the two operators are not
placed at the same spatial coordinates, is also given. In the current case,
the presence of the defect complicates the spin structure, making the problem
more difficult to handle. However the equations (24) are labeled by two
parameters $\bar{\Delta}$ and $n$, while in the case with no defect the
analogous equations are labeled by only one parameter, $n$. Therefore the set
of equations (24) is expected to be more constraining. Furthermore, numerical
techniques have been proposed to obtain predictions for the bulk one-point
functions of the $3d$ Ising model Iliesiu:2018zlz .121212In this case,
predictions are made by using the zero temperature CFT data as input. It would
be interesting to understand if the converse is also true, i.e., whether
finite temperature sum rules impose new constraints on the zero temperature
data El-Showk:2011yvt ; Marchetto:2023xap .

A few comments, concerning the sum rules (24), are in order before we conclude
this Section. First, it is important to notice that they correspond to a
restricted set of constraints, since we have used the KMS condition (18) in
the collinear limit (21). In Section 5, the set of equations (24) is checked
explicitly in the case of GFF. Second, there is a priori no positivity
condition on the OPE coefficients of (24). Note in particular that the
binomial coefficients take negative values for specific parameters.

###  3.2 Heavy operators

The sum rules derived in the previous Section are a consequence of the KMS
condition, and relate an infinite number of operators in different channels.
In particular, (18) relates light operators to heavy ones. At zero
temperature, this results into a relation between the lightest operator, the
defect identity, and the heavy operators Qiao:2017xif ; Fitzpatrick:2012yx ;
Komargodski:2012ek . Similar relations also exist at finite temperature
without defects Marchetto:2023xap . In this Section, we show how to extend
these relations to our setup with a line defect wrapping the thermal circle.

Let us consider again the collinear limit, in which the two scalar fields in
the bulk are placed on the same thermal circle. In this case, we have

$\langle\,\phi(\tau,\vec{x})\phi(0,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\sum_{{\mathcal{O}},{\widehat{\mathcal{O}}}}\mu_{{\mathcal{O}}\phi\phi}\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}\widehat{b}_{{\widehat{\mathcal{O}}}}\frac{\tau^{\Delta-2\Delta_{\phi}}}{|\vec{x}|^{\Delta}}z^{{\widehat{\Delta}}}\,.$
(25)

Notice that the defect identity certainly contributes to the bulk OPE, as the
two scalars are identical and the projection of the bulk identity to the
defect is simply $\mathds{1}\to\widehat{\mathds{1}}$. Therefore, we can select
the operators $\mathcal{O}$ (exchanged in the bulk channel) and
${\widehat{\mathcal{O}}}$ (exchanged in the defect channel) such that
$\Delta={\widehat{\Delta}}$ by picking the zeroth order in $|\vec{x}|$ in the
two-point function. This results in

$f_{\beta}(\tau)=\left.\langle\,\phi(\tau,\vec{x})\phi(0,\vec{x}){\mathcal{P}}\,\rangle_{\beta}\right|_{|\vec{x}|^{0}}=\sum_{\Delta={\widehat{\Delta}}}\mu_{{\mathcal{O}}\phi\phi}\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}\frac{\tau^{\Delta-2\Delta_{\phi}}}{\beta^{\Delta}}\,.$

The KMS condition (including parity $\tau\to-\tau$) now reads

$f_{\beta}(\tau)=f_{\beta}(\beta-\tau)\,.$ (26)

We can define an OPE density specific to this problem

$\rho(\widetilde{\Delta})=\sum_{\Delta}a_{\Delta}\delta(\widetilde{\Delta}-\Delta)\,,$
(27)

where
$a_{\Delta}=\mu_{{\mathcal{O}}\phi\phi}\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}$.
In the limit $\tau\to\beta$, the KMS equation for $f_{\beta}(\tau)$ becomes

$\int_{0}^{\infty}\mathrm{d}\Delta\,\rho(\Delta)\frac{\tau^{\Delta-2\Delta_{\phi}}}{\beta^{\Delta}}\sim\frac{1}{(\beta-\tau)^{2\Delta_{\phi}}}\,,$
(28)

with the right-hand side of the equation being the projection to the identity
contribution. From the inverse Laplace transform, one concludes that

$\rho(\Delta)\overset{\Delta\to\infty}{\sim}\frac{\Delta^{2\Delta_{\phi}-1}}{\Gamma(2\Delta_{\phi})}\left[1+{\mathcal{O}}\left(\frac{1}{\Delta}\right)\right]\,.$
(29)

Note that this result can also be obtained by using a suitable Ansatz
$\rho(\Delta)=A\Delta^{\alpha}$ for the OPE density.

Derivations of this type are common throughout the physics literature, as a
consequence of the channel duality (see e.g. Cardy:1986ie ; Fitzpatrick:2012yx
; Komargodski:2012ek ). It should be pointed out that it is not per se
mathematically precise, as it can happen that the OPE density oscillates
between two polynomial functions for high scaling dimensions (see Fig. 1 in
Qiao:2017xif ). In practice, this can only happen in case of a fine tuning
between the parameters $a_{\Delta}$.131313The most standard example in the
mathematical literature is discussed in (3.12) in Marchetto:2023xap . We refer
to this paper for more technical details. This issue can be circumvented by
either proving or assuming the positivity of the coefficients $a_{\Delta}$ at
large $\Delta$. In this case, (29) follows as a consequence of Tauberian
theorems, which were introduced in physics in Vladimirov:1978xx , and found
applications in various branches of theoretical physics Pappadopulo:2012jk ;
Das:2017vej ; Das:2020uax ; Pal:2020wwd ; Mukhametzhanov:2020swe ;
Mukhametzhanov:2019pzy . It would be interesting to make the derivation of
(29) rigorous by proving the positivity of $a_{\Delta}$ at large $\Delta$.

Finally, (29) should formally be understood as an average, since, for
instance, the OPE density cannot grow continuously in the case of a discrete
spectrum. In fact, the correct statement is rather

$\int_{0}^{\Delta}\mathrm{d}\widetilde{\Delta}\
\rho(\widetilde{\Delta})\overset{\Delta\to\infty}{\sim}\frac{\Delta^{2\Delta_{\phi}}}{\Gamma(2\Delta_{\phi}+1)}\left[1+{\mathcal{O}}\left(\frac{1}{\Delta}\right)\right]\,.$
(30)

##  4 Defect thermodynamics

We now discuss the thermodynamics of defects at finite temperature. We
determine the free energy and the entropy density of the system in terms of
OPE data, starting from the one-point function of the stress-energy tensor in
the presence of a line defect. To conclude, we discuss the free energy of
moving quarks in gauge theories.

###  4.1 Free energy density and entropy

Among all bulk operators, a special role is played by the stress-energy
tensor. It can be related to standard thermodynamic quantities, such as the
free energy and the entropy of the system. As a starting point, let us
consider the generic constraints arising from the preserved symmetry under
spatial rotations. At zero temperature, its one-point function in the presence
of a conformal line defect is fixed up to a single function of the marginal
couplings Kapustin:2005py . At finite temperature, the one-point function is
fixed in terms of two functions of the dimensionless ratio $z$ defined in (8)
and the marginal couplings. Employing $\mathrm{SO}(d-1)$-invariance together
with the tracelessness condition we can write the Ansatz for $d>2$

$\langle\,T^{ij}(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\frac{f_{1}(z)}{|\vec{x}|^{d}}\delta^{ij}-2\frac{x^{i}x^{j}}{|\vec{x}|^{d+2}}f_{2}(z)\,,$
(31)
$\langle\,T^{00}(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=-\frac{(d-1)f_{1}(z)-2f_{2}(z)}{|\vec{x}|^{d}}\,,\quad\langle\,T^{i0}(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=0\,.$
(32)

The case $d=2$ is discussed in Appendix C. There is no additional constraint
arising from symmetry. However, expanding the stress-energy tensor in the OPE
regime is possible. For instance, the time-time component yields the equation

$-(d-1)f_{1}(z)+2f_{2}(z)=\sum_{{\widehat{\mathcal{O}}}}\lambda_{T{\widehat{\mathcal{O}}}}\,\widehat{b}_{\widehat{\mathcal{O}}}\,z^{{\widehat{\Delta}}}\,.$
(33)

It is important to notice that $f_{1}(z)$ and $f_{2}(z)$ are not independent
functions. The (broken) Ward identities associated with spatial translations
can be expressed as141414See Appendix A for a derivation.

$\partial_{\mu}\langle T^{\mu
j}(\tau,\vec{x}){\mathcal{P}}\rangle_{\beta}=\delta^{(d-1)}(\vec{x})D^{j}(\tau)\,,$
(34)

which can be used for $|\vec{x}|\neq 0$ to relate $f_{1}(z)$ and $f_{2}(z)$
via the differential equation (for $z>0$)

$-d\,f_{1}(z)+4f_{2}(z)+z(f_{1}^{\prime}(z)-2f_{2}^{\prime}(z))=0\,.$ (35)

This leads to the following analytical solution for $f_{2}(z)$:

$f_{2}(z)=c_{1}z^{2}+z^{2}\int_{0}^{z}\mathrm{d}y\
\frac{-d\,f_{1}(y)+yf_{1}^{\prime}(y)}{2y^{3}}$ (36)

where $c_{1}$ is an integration constant.

Combining the equation (35) with the OPE expansion (33) gives two equations
for $f_{1},f_{2}$ in terms of the data
$\lambda_{T{\widehat{\mathcal{O}}}}\,\widehat{b}_{\widehat{\mathcal{O}}}$ and
$\widehat{\Delta}$

$\begin{split}&f_{1}(z)=\sum_{{\widehat{\mathcal{O}}}}\lambda_{T{\widehat{\mathcal{O}}}}\,\widehat{b}_{\widehat{\mathcal{O}}}\,\frac{2-{\widehat{\Delta}}}{(d-2)({\widehat{\Delta}}-1)}z^{{\widehat{\Delta}}}\,,\\\
&f_{2}(z)=\sum_{{\widehat{\mathcal{O}}}}\frac{\lambda_{T{\widehat{\mathcal{O}}}}\,\widehat{b}_{\widehat{\mathcal{O}}}}{2}\left(1+\frac{d-1}{d-2}\frac{2-{\widehat{\Delta}}}{{\widehat{\Delta}}-1}\right)z^{{\widehat{\Delta}}}\,.\end{split}$
(37)

Asymptotic for $f_{1}$ and $f_{2}$ are given by

$f_{1}(z)\overset{z\to
0}{\sim}\frac{2}{d-2}\lambda_{T\widehat{\mathds{1}}}\,,\quad
f_{2}(z)\overset{z\to
0}{\sim}-\frac{d/2}{d-2}\lambda_{T\widehat{\mathds{1}}}\,,\quad
f_{1}(z)\overset{z\to\infty}{\sim}\frac{b_{T}}{d}z^{d}\,,\quad
f_{2}(z)\overset{z\to\infty}{\sim}0\,,$ (38)

where $\lambda_{T\widehat{\mathds{1}}}$ is the defect one-point function of
the stress-energy tensor (at zero temperature), while $b_{T}$ is the thermal
one-point function of the defect (in absence of the defect). The first two
equations in (38) correspond to the zero temperature limit, while the other
two should be understood as the limit in which the defect is far away from the
operators or equivalently the high temperature limit, thus reproducing the
bulk thermal CFT.

The main motivation for studying the stress-energy tensor is its direct
relation to the energy density of the system151515The minus sign is
conventional (see Iliesiu:2018fao ; Marchetto:2023xap ). It is there to ensure
that the energy density is exactly equal to the stress-energy tensor one-point
function after the appropriate Wick rotation.

$E(\vec{x})=\frac{{\mathcal{E}}(z)}{|\vec{x}|^{d}}=-\langle\,T^{00}(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}\,.$
(39)

Since the system that we are considering is at finite temperature, it is
possible to consider the thermodynamic equation

$F(\vec{x})=E(\vec{x})-TS(\vec{x})=E(\vec{x})+T\frac{\mathrm{d}F(\vec{x})}{\mathrm{d}T}\,,$
(40)

which can be used to derive the free energy density $F(\vec{x})$ and the
entropy density $S(\vec{x})$. We define the dimensionless quantity
$\mathfrak{f}(z)$ through $F(\vec{x})=|\vec{x}|^{-d}\mathfrak{f}(z)$, and (40)
can be recast as

$\mathfrak{f}(z)={\mathcal{E}}(z)+z\mathfrak{f}^{\prime}(z)\,.$ (41)

This is solved by

$\mathfrak{f}(z)=c_{2}z-z\int_{1}^{z}\frac{\mathrm{d}y}{y^{2}}\
{\mathcal{E}}(y)\ ,$ (42)

where $c_{2}$ is an integration constant. The entropy density can be
calculated by applying a derivative with respect to the temperature

$S(\vec{x})=-\frac{\mathrm{d}F}{\mathrm{d}T}=-\frac{1}{|\vec{x}|^{d-1}}\left(c_{2}-\int_{1}^{z}\frac{\mathrm{d}y}{y^{2}}{\mathcal{E}}(y)-\frac{{\mathcal{E}}(z)}{z}\right)\,.$
(43)

In terms of CFT data,

$F(\vec{x})=\frac{1}{|\vec{x}|^{d}}\left(c_{3}z+\sum_{{\widehat{\mathcal{O}}}}\lambda_{T{\widehat{\mathcal{O}}}}\,\widehat{b}_{\widehat{\mathcal{O}}}\,\frac{1}{{\widehat{\Delta}}-1}z^{{\widehat{\Delta}}}\,\right)$
(44)

where $c_{3}$ is a function of CFT data to be determined below. The entropy
density is

$S(\vec{x})=-\frac{1}{|\vec{x}|^{d-1}}\left(c_{3}+\sum_{{\widehat{\mathcal{O}}}}\lambda_{T{\widehat{\mathcal{O}}}}\,\widehat{b}_{\widehat{\mathcal{O}}}\,\frac{{\widehat{\Delta}}}{{\widehat{\Delta}}-1}z^{{\widehat{\Delta}}-1}\right)$
(45)

To fix $c_{3}$ we need to know the defect entropy at zero temperature. We
notice that this limit requires that there are no defect operators (other than
the identity) with ${\widehat{\Delta}}\leq 1$ and
$\lambda_{T{\widehat{\mathcal{O}}}}\,\widehat{b}_{\widehat{\mathcal{O}}}\neq
0$, since otherwise the zero temperature limit is diverging. This condition
can be explicitly verified: we perform this analysis in Appendix D. Finally,
when the zero temperature limit of the entropy vanishes, then $c_{3}$ can be
set to zero.

###  4.2 The free energy of a quark

When considering a Polyakov loop in gauge theory, its expectation value is
known to be an order parameter for the confinement/deconfinement transition
Polyakov:1975rs . All the discussions in this paper can be easily adapted to
the case of a Polyakov loop, but paying attention to the definition and
normalization of the loop. We are interested in providing a definition for
Polyakov lines at finite temperature which is able to reflect their role as
order parameters. Let us consider an $\mathrm{SU}(N)$ gauge theory where we
define the operator

$P=\frac{1}{N}\operatorname{Tr}\mathrm{P}\exp\left(\int_{0}^{\beta}\mathrm{d}\tau\
A_{0}(\tau)\right)\ ,$ (46)

where $\mathrm{P}$ represent the path order. The one-point function of the
operator above is naturally divergent, therefore the expression (46) needs to
be regularized. The most common regularization employed for conformal defects
is to normalize any correlation function with the expectation value of the
defect itself

$\langle\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n}){\mathcal{P}}\rangle=\frac{\langle\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})P\rangle}{\langle
P\rangle}\ .$ (47)

In this paper we make use of this type of normalization. Nonetheless in this
convention the finite temperature expectation value
$\langle{\mathcal{P}}\rangle_{\beta}$ is identically equal to the zero
temperature vacuum expectation value $\langle{\mathcal{P}}\rangle=1$, in
contrast with the requirement for Polyakov loops expectation value to be
sensitive to the free energy of the quark $F_{\text{quark}}$

$\langle{\mathcal{P}}\rangle_{\beta}=e^{-\beta\left(F_{\text{quark}}-F_{\text{free
quark}}\right)}\ ,$ (48)

where $F_{\text{free quark}}$ is the free energy of a free quark. To
compensate for this discrepancy it is possible to define

${\mathcal{P}}=\frac{\operatorname{Tr}\mathrm{P}\exp\left(\int_{0}^{\beta}\mathrm{d}\tau\
A_{0}(\tau)\right)}{N\langle P_{\text{free}}\rangle_{\beta}}\ ,$ (49)

where the denominator is the expectation value of the Polyakov loop (46) at
zero couplings. The definition above matches the expectation (48) by
construction and makes it a well defined order parameter for the
confinement/deconfinement transition: in fact in the non confining phase
$F_{\text{quark}}-F_{\text{free quark}}$ is expected to be finite and
therefore $\langle{\mathcal{P}}\rangle_{\beta}$ is expected to be finite,
while in the confining phase $F_{\text{quark}}-F_{\text{free quark}}$ is
expected to be infinite and therefore $\langle{\mathcal{P}}\rangle_{\beta}=0$.
The limitation of this regularization is that a notion of $P_{\text{free}}$ is
needed and therefore a Lagrangian description is required. In this paper we
are interested in the most general setup and we will therefore use
${\mathcal{P}}$ for the regularized defect insertion.

As already mentioned above with the proper normalization of the Polyakov loop,
typical to Lagrangian gauge theories, the expectation value of the loop itself
is expected to be the exponential of
$-\beta\left(F_{\text{quark}}-F_{\text{free quark}}\right)$. Observe that the
expectation value of the loop is the partition function of the full theory
divided by the partition function in absence of the Polyakov loop. In terms of
free energies

$F_{\text{quark}}-F_{\text{free
quark}}=\int\mathrm{d}^{d-1}x\left[F(\vec{x})-F_{\text{free}}(\vec{x})-\frac{b_{T}}{d}\right]\
,$ (50)

where $b_{T}$ is the thermal one-point function of the stress-energy tensor
and represents the finite temperature contribution to the free energy.

##  5 Applications

We present here applications of the concepts developed in the previous
Sections for simple models, in which the computations can be carried out
analytically. The first example is the case of generalized free fields, while
the second one is the (interacting) $\mathrm{O}(N)$ model in the
$\varepsilon$-expansion and at large $N$.

###  5.1 Magnetic lines in generalized free field theory

To begin, we study the simple case of a magnetic line in Generalized Free
Field (GFF) theory, in any spacetime dimension $d$. The thermal defect CFT is
defined via the action

$S=S_{\text{GFF}}+h\int_{0}^{\beta}\mathrm{d}\tau\ \phi(\tau,\vec{0})\,,$ (51)

where we denote as $\phi$ the fundamental field of the GFF theory and as $h$
the coupling constant of the defect. The defect in (51) is conformal if
$\Delta_{\phi}=1$, which corresponds to non-local theories in $2\leq d<4$ and
a free scalar field theory in $d=4$. Since the theory is free, it is possible
to write the exact scalar propagator at finite temperature by using the method
of images. We represent it as a Feynman diagrams using a solid black line

$\displaystyle=\langle\,\phi(\tau,\vec{x})\phi(0,\vec{0})\,\rangle_{\beta}$
$\displaystyle=\frac{\pi}{2\beta|\vec{x}|}\left[\coth\left(\frac{\pi}{\beta}(|\vec{x}|-i\tau)\right)+\coth\left(\frac{\pi}{\beta}(|\vec{x}|+i\tau)\right)\right]\,.$
(52)

Finally, the defect itself is defined as

${\mathcal{P}}=\frac{1}{\langle\,{\mathcal{P}}\,\rangle}_{\beta}\exp\left[-h\int_{0}^{\beta}\mathrm{d}\tau\
\phi(\tau,\vec{0})\right]\,.$ (53)

In Feynman diagrams, we represent the defect with a red line. The
corresponding Feynman rule is defined as

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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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=-h\int_{0}^{\beta}\mathrm{d}\tau\,.$ (54)

####  5.1.1 One-point functions of bulk operators

We now compute the one-point functions of the bulk operators $\phi$ and
$\phi^{2}$. This can be done straightforwardly by considering the Feynman
diagrams of free theory.

##### One-point function of $\phi$.

The one-point function of $\phi$ consists of a single diagram161616There is in
principle an additional disconnected term, corresponding to the thermal one-
point function of $\phi$. However, this term vanishes due to the
$\mathbb{Z}_{2}$ symmetry of GFF in absence of the defect.

$\langle\,\phi(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\leavevmode\hbox
to36.27pt{\vbox to29.45pt{\pgfpicture\makeatletter\hbox{\hskip
4.76773pt\lower-14.72638pt\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{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{7.8571pt}{2.35703pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-2.35703pt}{14.22638pt}{-4.26773pt}{7.8571pt}{-4.26773pt}{0.0pt}\pgfsys@curveto{-4.26773pt}{-7.8571pt}{-2.35703pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{2.35703pt}{-14.22638pt}{4.26773pt}{-7.8571pt}{4.26773pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{31.2982pt}{0.0pt}\pgfsys@curveto{31.2982pt}{1.5715pt}{30.02426pt}{2.84544pt}{28.45276pt}{2.84544pt}\pgfsys@curveto{26.88126pt}{2.84544pt}{25.60732pt}{1.5715pt}{25.60732pt}{0.0pt}\pgfsys@curveto{25.60732pt}{-1.5715pt}{26.88126pt}{-2.84544pt}{28.45276pt}{-2.84544pt}\pgfsys@curveto{30.02426pt}{-2.84544pt}{31.2982pt}{-1.5715pt}{31.2982pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@moveto{5.69044pt}{0.0pt}\pgfsys@curveto{5.69044pt}{0.78575pt}{5.05348pt}{1.42271pt}{4.26773pt}{1.42271pt}\pgfsys@curveto{3.48198pt}{1.42271pt}{2.84502pt}{0.78575pt}{2.84502pt}{0.0pt}\pgfsys@curveto{2.84502pt}{-0.78575pt}{3.48198pt}{-1.42271pt}{4.26773pt}{-1.42271pt}\pgfsys@curveto{5.05348pt}{-1.42271pt}{5.69044pt}{-0.78575pt}{5.69044pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@fillstroke\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}}\,,$ (55)

The empty circle on the right depicts the insertion of the operator
$\phi(\tau,\vec{x})$ in the bulk. By inserting (52), it is easy to show that
the diagram (55) exactly corresponds to the zero temperature one-point
function of $\phi$, in the presence of the magnetic line

$\displaystyle=-h\int_{0}^{\beta}\mathrm{d}\tilde{\tau}\
\langle\,\phi(\tau,\vec{x})\phi(\tilde{\tau})\,\rangle_{\beta}=-\frac{h\pi}{|\vec{x}|}\,.$
(56)

By using the inversion formula (17) (or by simply comparing (56) to (9)), we
find that only the defect identity $\widehat{\mathds{1}}$ contributes to the
defect OPE in this correlator. In fact, by using the equation of motion of
$\phi$ in the free scalar theory in $4d$, it is possible to show that the only
operators appearing in the defect expansion of $\phi$ are of the type
${\widehat{\mathcal{O}}}_{{\widehat{\Delta}}}=\partial^{i_{1}}\ldots\partial^{i_{s}}\widehat{\phi}(\tau)$
(i.e. ${\widehat{\Delta}}=s+1$). Since operators on the defect are, in the
free case, restrictions of operators in the bulk, it is expected that
$\langle\,\partial^{i_{1}}\ldots\partial^{i_{s}}\widehat{\phi}(\tau){\mathcal{P}}\,\rangle_{\beta}=0$,
in agreement with the result in equation (56). In the case of a GFF the most
general form of the operators appearing in the bulk to defect also includes
for instance $\partial^{i_{1}}\ldots\partial^{i_{s}}\Box^{k}\phi$. Those
operators share conformal dimensions with the operators discussed above, but
they have different transverse spin: the conclusion in the GFF is that the sum
over one-point functions of operators with the same conformal dimensions is
zero.

##### One-point function of $\phi^{2}$.

It is also straightforward to calculate the one-point function of $\phi^{2}$.
However, in this case there is a second diagram corresponding to the thermal
one-point function $\langle\,\phi^{2}\,\rangle_{\beta}$

$\langle\,\phi^{2}(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\leavevmode\hbox
to36.27pt{\vbox to29.45pt{\pgfpicture\makeatletter\hbox{\hskip
4.76773pt\lower-14.72638pt\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{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{7.8571pt}{2.35703pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-2.35703pt}{14.22638pt}{-4.26773pt}{7.8571pt}{-4.26773pt}{0.0pt}\pgfsys@curveto{-4.26773pt}{-7.8571pt}{-2.35703pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{2.35703pt}{-14.22638pt}{4.26773pt}{-7.8571pt}{4.26773pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@curveto{28.45276pt}{3.92854pt}{25.26811pt}{7.11319pt}{21.33957pt}{7.11319pt}\pgfsys@curveto{17.41103pt}{7.11319pt}{14.22638pt}{3.92854pt}{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-3.92854pt}{17.41103pt}{-7.11319pt}{21.33957pt}{-7.11319pt}\pgfsys@curveto{25.26811pt}{-7.11319pt}{28.45276pt}{-3.92854pt}{28.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{31.2982pt}{0.0pt}\pgfsys@curveto{31.2982pt}{1.5715pt}{30.02426pt}{2.84544pt}{28.45276pt}{2.84544pt}\pgfsys@curveto{26.88126pt}{2.84544pt}{25.60732pt}{1.5715pt}{25.60732pt}{0.0pt}\pgfsys@curveto{25.60732pt}{-1.5715pt}{26.88126pt}{-2.84544pt}{28.45276pt}{-2.84544pt}\pgfsys@curveto{30.02426pt}{-2.84544pt}{31.2982pt}{-1.5715pt}{31.2982pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fillstroke\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
to36.27pt{\vbox to29.45pt{\pgfpicture\makeatletter\hbox{\hskip
4.76773pt\lower-14.72638pt\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{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{7.8571pt}{2.35703pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-2.35703pt}{14.22638pt}{-4.26773pt}{7.8571pt}{-4.26773pt}{0.0pt}\pgfsys@curveto{-4.26773pt}{-7.8571pt}{-2.35703pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{2.35703pt}{-14.22638pt}{4.26773pt}{-7.8571pt}{4.26773pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{7.11319pt}\pgfsys@lineto{28.45276pt}{1.42271pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{-1.42271pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{31.2982pt}{0.0pt}\pgfsys@curveto{31.2982pt}{1.5715pt}{30.02426pt}{2.84544pt}{28.45276pt}{2.84544pt}\pgfsys@curveto{26.88126pt}{2.84544pt}{25.60732pt}{1.5715pt}{25.60732pt}{0.0pt}\pgfsys@curveto{25.60732pt}{-1.5715pt}{26.88126pt}{-2.84544pt}{28.45276pt}{-2.84544pt}\pgfsys@curveto{30.02426pt}{-2.84544pt}{31.2982pt}{-1.5715pt}{31.2982pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{3.69899pt}{7.11319pt}\pgfsys@moveto{5.1217pt}{7.11319pt}\pgfsys@curveto{5.1217pt}{7.89894pt}{4.48474pt}{8.5359pt}{3.69899pt}{8.5359pt}\pgfsys@curveto{2.91324pt}{8.5359pt}{2.27628pt}{7.89894pt}{2.27628pt}{7.11319pt}\pgfsys@curveto{2.27628pt}{6.32744pt}{2.91324pt}{5.69048pt}{3.69899pt}{5.69048pt}\pgfsys@curveto{4.48474pt}{5.69048pt}{5.1217pt}{6.32744pt}{5.1217pt}{7.11319pt}\pgfsys@closepath\pgfsys@moveto{3.69899pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{3.69899pt}{-7.11319pt}\pgfsys@moveto{5.1217pt}{-7.11319pt}\pgfsys@curveto{5.1217pt}{-6.32744pt}{4.48474pt}{-5.69048pt}{3.69899pt}{-5.69048pt}\pgfsys@curveto{2.91324pt}{-5.69048pt}{2.27628pt}{-6.32744pt}{2.27628pt}{-7.11319pt}\pgfsys@curveto{2.27628pt}{-7.89894pt}{2.91324pt}{-8.5359pt}{3.69899pt}{-8.5359pt}\pgfsys@curveto{4.48474pt}{-8.5359pt}{5.1217pt}{-7.89894pt}{5.1217pt}{-7.11319pt}\pgfsys@closepath\pgfsys@moveto{3.69899pt}{-7.11319pt}\pgfsys@fillstroke\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}}\,.$ (57)

This expression can be made finite by considering the appropriate normal
ordering. The value of the thermal one-point function can be found in Appendix
C in Marchetto:2023fcw  or in the discussion on GFF in Iliesiu:2018fao . This
leads to

$\langle\,\phi^{2}(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\langle\,\phi^{2}(\tau,\vec{x}){\mathcal{P}}\,\rangle+\langle\,\phi^{2}\,\rangle_{\beta}=\frac{1}{|\vec{x}|^{2}}\left(h^{2}\pi^{2}+\frac{\pi^{2}}{6}z^{2}\right)\,,$
(58)

where the first term can be calculated in the same way as (56). The
expectation value independent from $\beta$ indicates the zero temperature
correlator. We conclude that the defect operators contributing to the one-
point function of $\phi^{2}$ are the defect identity $\widehat{\mathds{1}}$
and a defect operator of dimension ${\widehat{\Delta}}=2$. In GFF, since we
choose $\Delta_{\phi}=1$, the only two defect operators of dimension
$\widehat{\Delta}=2$ are $\widehat{\phi}^{\,\,2}$ and the displacement
operator $D^{i}=\partial^{i}\phi$. $\mathrm{SO}(d-1)$ symmetry prevents the
latter from gaining a non-zero one-point function, and therefore we identify
the second contribution in (58) as the one-point function of
$\widehat{\phi}^{\,\,2}$ with defect one-point function

$\langle\,\widehat{\phi}^{\,\,2}\,\rangle_{\beta}=\langle\,\phi^{2}\,\rangle_{\beta}=\frac{\pi^{2}}{6\beta^{2}}\,,$
(59)

as expected.

####  5.1.2 The two-point function
$\langle\,\phi\phi{\mathcal{P}}\,\rangle_{\beta}$

The two-point function of two elementary scalar fields $\phi$ in presence of
the magnetic line reads

$\displaystyle\langle\,\phi(\tau_{1},\vec{x}_{1})\phi(\tau_{2},\vec{x}_{2}){\mathcal{P}}\,\rangle_{\beta}$
$\displaystyle=\leavevmode\hbox to36.27pt{\vbox
to29.45pt{\pgfpicture\makeatletter\hbox{\hskip 4.76773pt\lower-14.72638pt\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{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{7.8571pt}{2.35703pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-2.35703pt}{14.22638pt}{-4.26773pt}{7.8571pt}{-4.26773pt}{0.0pt}\pgfsys@curveto{-4.26773pt}{-7.8571pt}{-2.35703pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{2.35703pt}{-14.22638pt}{4.26773pt}{-7.8571pt}{4.26773pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{7.11319pt}\pgfsys@lineto{28.45276pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{7.11319pt}\pgfsys@moveto{31.2982pt}{7.11319pt}\pgfsys@curveto{31.2982pt}{8.6847pt}{30.02426pt}{9.95863pt}{28.45276pt}{9.95863pt}\pgfsys@curveto{26.88126pt}{9.95863pt}{25.60732pt}{8.6847pt}{25.60732pt}{7.11319pt}\pgfsys@curveto{25.60732pt}{5.54169pt}{26.88126pt}{4.26775pt}{28.45276pt}{4.26775pt}\pgfsys@curveto{30.02426pt}{4.26775pt}{31.2982pt}{5.54169pt}{31.2982pt}{7.11319pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
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$\displaystyle=\langle\,\phi(\tau_{1},\vec{x}_{1})\phi(\tau_{2},\vec{x}_{2})\,\rangle_{\beta}+\langle\,\phi(\tau_{1},\vec{x}_{1}){\mathcal{P}}\,\rangle_{\beta}\langle\,\phi(\tau_{2},\vec{x}_{2}){\mathcal{P}}\,\rangle_{\beta}\,.$
(60)

It can be seen from equations (55) and (56) that the second term is
independent of $\beta$, and thus all the thermal effects are contained in the
disconnected term, corresponding to the two-point function without defect.

We now consider the collinear limit discussed in Section 3.1. In this case,
the disconnected term can be deduced from 52 to be

$\langle\,\phi(\tau,\vec{x})\phi(0,\vec{x})\,\rangle_{\beta}=\frac{\pi^{2}}{\beta^{2}}\csc^{2}\left(\frac{\pi\tau}{\beta}\right)=\frac{2}{\beta^{2}}\sum_{k=0}^{\infty}(2k-1)\zeta_{2k}\left(\frac{\tau}{\beta}\right)^{2k-2}\,.$
(61)

where $\zeta_{2k}$ is the Riemann $\zeta$-function evaluated at $2k$. This
leads to the expression

$\langle\,\phi(\tau,\vec{x})\phi(0,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\langle\,\phi(\tau,\vec{x})\phi(0,\vec{x}){\mathcal{P}}\,\rangle+\frac{1}{\beta^{2}}\sum_{k=1}^{\infty}c_{k}\left(\frac{\tau}{\beta}\right)^{2k-2}\,,$
(62)

where the first term is the zero temperature expression, while
$c_{k}=2(2k-1)\zeta_{2k}$. This should be compared to the OPE expansion (5)

$\langle\,\phi(\tau,\vec{x})\phi(0,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\langle\,\phi(\tau,\vec{x})\phi(0,\vec{x}){\mathcal{P}}\,\rangle+\sum_{{\mathcal{O}}\neq\mathds{1}}\sum_{{\widehat{\Delta}}>0}\mu_{\phi\phi{\mathcal{O}}}\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}\widehat{b}_{\widehat{\mathcal{O}}}\frac{\tau^{\Delta-2}}{\beta^{\widehat{\Delta}}|x|^{\Delta-{\widehat{\Delta}}}}\,.$
(63)

The interpretation is that, from the operators appearing in this OPE, only the
ones satisfying $\Delta={\widehat{\Delta}}$ receive a finite temperature
correction. More explicitly,

$\mu_{\phi\phi{\mathcal{O}}}\lambda_{{\mathcal{O}}{\widehat{\mathcal{O}}}}\widehat{b}_{\widehat{\mathcal{O}}}=\begin{cases}c_{\Delta}\,,&\text{if
}\Delta={\widehat{\Delta}}\neq 0\text{ and even}\,,\\\
\mu_{\phi\phi{\mathcal{O}}}\lambda_{{\mathcal{O}}\widehat{\mathds{1}}}\,,&\text{if
}{\widehat{\Delta}}=0\,,\\\ 0\,,&\text{otherwise}\,.\end{cases}$ (64)

It is straightforward to check analytically that these explicit results
satisfy (23) for all odd $n$, providing a nice consistency check. This boils
down to the fact that all odd derivatives of $\text{csc}^{2}(x)$ at $x=\pi/2$
vanish.

We notice that in the two-point functions above, in the case of the free
scalar theory in $4d$, the non-vanishing and $\beta$-dependent terms appear
corresponding to the operators
$\widehat{\phi}\,\partial^{i_{1}}\ldots\partial^{i_{s}}\widehat{\phi}$, which
naturally appear in the bulk-to-defect OPEs of the bulk operators in the OPE
$\widehat{\phi}\times\widehat{\phi}$. In the case of a GFF we have to add also
the operators which vanish in the free theory because of equations of motions
which are for instance
$\widehat{\phi}\,\partial^{i_{1}}\ldots\partial^{i_{s}}\Box^{k}\widehat{\phi}$.
As commented above, all those operators are degenerate in the conformal
dimensions and differ for the value of the transverse spin. Therefore in the
limit in which the two operators are at the same distance from the defect,
since we are not sensitive to the transverse spin, the $\beta$-dependence is
due to sum over operators of the same conformal dimensions.

####  5.1.3 Free energy and entropy

The one-point function of the stress-energy tensor can also be computed
explicitly when the theory has a well-defined stress-energy tensor: this is
again the case of the free scalar theory in $d=4$.

The result for the free scalar theory is

$\langle\,T^{\mu\nu}{\mathcal{P}}\,\rangle_{\beta}=\langle\,T^{\mu\nu}{\mathcal{P}}\,\rangle+\langle\,T^{\mu\nu}\,\rangle_{\beta}\,.$
(65)

Specializing to the case $\mu=\nu=0$, we have

$\langle\,T^{00}{\mathcal{P}}\,\rangle_{\beta}=-\frac{1}{|\vec{x}|^{4}}\left(\frac{h^{2}\pi^{2}}{2}+\frac{2\pi^{2}}{15}z^{4}\right)\,.$
(66)

The value of the thermal one-point function of the stress-energy tensor for
the free scalar theory in four dimensions is given, e.g., in Marchetto:2023fcw
and Iliesiu:2018fao . As explained in Section 4.1, the one-point function of
the stress-energy tensor is related to the energy density, the free energy
density and the entropy density of the theory. By using (42) and (43), we find

$F(\vec{x})=\frac{1}{|\vec{x}|^{4}}\biggl{(}c_{2}z+\frac{\pi^{2}}{90}(1-z)(45h^{2}+4z(1+z+z^{2}))\biggr{)}\,.$
(67)

Differentiating with respect to the temperature, we obtain the entropy density
of the system

$S(\vec{x})=\frac{s(\vec{x})}{|\vec{x}|^{3}}=\frac{\pi^{2}\left(45h^{2}+16z^{3}-4\right)-90c_{2}}{90|\vec{x}|^{3}}\,.$
(68)

The limit $z\to 0$ produces

$s(z)\overset{z\to
0}{\sim}\frac{\pi^{2}\left(45h^{2}-4\right)-90c_{2}}{90}\,,$ (69)

which is expected to be zero when we require the entropy to vanish at zero
temperature. This condition fixes $c_{2}$, and we end up with

$F(\vec{x})=\frac{\pi^{2}\left(45h^{2}-4z^{4}\right)}{90|\vec{x}|^{4}}\,,\quad
S(\vec{x})=\frac{8\pi^{2}T^{3}}{45}\,.$ (70)

Observe that, in the free theory, the entropy density does not depend on the
defect. We do not expect this to carry on to the more general case of
interacting theories, where terms involving the temperature and the defect
coupling should mix. Moreover we observe that the free energy blows up at
$|\vec{x}|\to 0$, as expected. Furthermore, it is a monotonic function of $z$
and its limit when $z\to\infty$ reproduces the free energy of the theory in
the absence of the defect.

###  5.2 The magnetic line in the $\mathrm{O}(N)$ model

We now turn our attention to the $\mathrm{O}(N)$ model, to provide an example
in which interactions are turned on, while computations can still be carried
out analytically. At zero temperature, this theory was extensively studied
without defect insertions Kos:2016ysd ; Dey:2016zbg ; Rychkov:2018vya ;
Giombi:2019upv ; Giombi:2020enj , and the theory coupled to a localized
magnetic line has been studied extensively Cuomo:2021kfm ; Gimenez-
Grau:2022ebb ; Gimenez-Grau:2022czc . At finite temperature, the
$\mathrm{O}(N)$ model without defects is one of the most studied thermal CFTs
Iliesiu:2018fao ; Petkou:2018ynm ; David:2023uya ; David:2024naf . In this
Section, we present two different but complementary approaches: the
$\varepsilon$-expansion and the large $N$ analysis.171717We are grateful to
Simone Giombi for interesting discussions regarding this Section.

####  5.2.1 $\varepsilon$-expansion

##### The Wilson-Fisher fixed point.

The $\mathrm{O}(N)$ model in $d$ dimensions, at finite temperature and in the
presence of a magnetic line, is described by the action

$S=\int_{0}^{\beta}\mathrm{d}\tau\ \int\mathrm{d}^{d-1}x\
\left[\frac{1}{2}(\partial_{\mu}\phi_{i})^{2}+\frac{\lambda}{4!}(\phi_{i}\phi_{i})^{2}+h\,\delta^{d-1}(\vec{x})\phi_{1}(\tau)\right]\,.$
(71)

The action corresponds to a conformal defect for specific values of the
coupling constants $\lambda$ and $h$. The idea of the $\varepsilon$-expansion
is to set $d=4-\varepsilon$ and find the fixed point perturbatively as a
function of $\varepsilon$. This fixed point is called the Wilson-Fisher fixed
point, for which the coupling constants are known to take the following form
Cuomo:2021kfm ; Gimenez-Grau:2022ebb

$\displaystyle\frac{\lambda_{\star}}{(4\pi)^{2}}$
$\displaystyle=\frac{3}{N+8}\varepsilon+\frac{9(3N+14)}{(N+8)^{3}}\varepsilon^{2}+\mathcal{O}(\varepsilon^{3})\,,$
(72) $\displaystyle h_{\star}^{2}$
$\displaystyle=N+8+\frac{4N^{2}+45N+170}{2N+16}\varepsilon+\mathcal{O}(\varepsilon^{2})$
(73)

such that the defect is conformal. Note that, at leading order,
$\lambda_{\star}\sim\varepsilon$ and $h_{\star}\sim 1$. The Feynman rule
associated to the quartic interaction in the bulk simply reads

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d^{d-1}x\,,$ (74)

while the thermal bulk propagator in arbitrary dimensions is

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=\frac{\Gamma(d/2-1)}{4\pi^{d/2}}\sum_{m\in\mathbb{Z}}\frac{1}{(|\vec{x}|^{2}+|\tau+m\beta|^{2})^{d/2-1}}\,.$
(75)

##### One-point function of $\phi_{1}$.

At order ${\mathcal{O}}(\varepsilon)$, the one-point function of $\phi_{1}$ is
given by the following Feynman diagrams

$\langle\,\phi_{1}(\tau,\vec{x}){\mathcal{P}}\,\rangle_{\beta}=\leavevmode\hbox
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}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{0.78575pt}{22.12532pt}{1.42271pt}{21.33957pt}{1.42271pt}\pgfsys@curveto{20.55382pt}{1.42271pt}{19.91685pt}{0.78575pt}{19.91685pt}{0.0pt}\pgfsys@curveto{19.91685pt}{-0.78575pt}{20.55382pt}{-1.42271pt}{21.33957pt}{-1.42271pt}\pgfsys@curveto{22.12532pt}{-1.42271pt}{22.76228pt}{-0.78575pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\mathcal{O}(\varepsilon^{2})\,.$
(76)

To compute these diagrams, we borrow the conventions for the normalization of
the vertices and of the propagators from Gimenez-Grau:2022ebb  and we make use
of the master integrals

$\displaystyle\int_{-\infty}^{\infty}\frac{\mathrm{d}\tau}{(|\vec{x}|^{2}+\tau^{2})^{1-\varepsilon/2}}=\frac{\sqrt{\pi}\Gamma\left(\frac{1-\varepsilon}{2}\right)}{\Gamma(1-\varepsilon/2)|\vec{x}|^{1-\varepsilon}}\,,$
(77)
$\displaystyle\int\frac{\mathrm{d}^{d-1}x_{2}}{|\vec{x}_{12}|^{a}|\vec{x}_{2}|^{b}}=\frac{\Gamma\left(\frac{d-1-a}{2}\right)\Gamma\left(\frac{d-1-b}{2}\right)\Gamma\left(\frac{a+b+1-d}{2}\right)}{\Gamma\left(\frac{a}{2}\right)\Gamma\left(\frac{b}{2}\right)\Gamma\left(\frac{2(d-1)-a-b}{2}\right)}\frac{\pi^{(d-1)/2}}{|\vec{x}_{1}|^{a+b+1-d}}\,.$
(78)

The first diagram is similar to (56)

$\leavevmode\hbox to36.27pt{\vbox
to29.45pt{\pgfpicture\makeatletter\hbox{\hskip 4.76773pt\lower-14.72638pt\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{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{7.8571pt}{2.35703pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-2.35703pt}{14.22638pt}{-4.26773pt}{7.8571pt}{-4.26773pt}{0.0pt}\pgfsys@curveto{-4.26773pt}{-7.8571pt}{-2.35703pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{2.35703pt}{-14.22638pt}{4.26773pt}{-7.8571pt}{4.26773pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{31.2982pt}{0.0pt}\pgfsys@curveto{31.2982pt}{1.5715pt}{30.02426pt}{2.84544pt}{28.45276pt}{2.84544pt}\pgfsys@curveto{26.88126pt}{2.84544pt}{25.60732pt}{1.5715pt}{25.60732pt}{0.0pt}\pgfsys@curveto{25.60732pt}{-1.5715pt}{26.88126pt}{-2.84544pt}{28.45276pt}{-2.84544pt}\pgfsys@curveto{30.02426pt}{-2.84544pt}{31.2982pt}{-1.5715pt}{31.2982pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@moveto{5.69044pt}{0.0pt}\pgfsys@curveto{5.69044pt}{0.78575pt}{5.05348pt}{1.42271pt}{4.26773pt}{1.42271pt}\pgfsys@curveto{3.48198pt}{1.42271pt}{2.84502pt}{0.78575pt}{2.84502pt}{0.0pt}\pgfsys@curveto{2.84502pt}{-0.78575pt}{3.48198pt}{-1.42271pt}{4.26773pt}{-1.42271pt}\pgfsys@curveto{5.05348pt}{-1.42271pt}{5.69044pt}{-0.78575pt}{5.69044pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@fillstroke\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{1}{4}h\pi^{\frac{\varepsilon-3}{2}}\Gamma\left(\frac{1-\varepsilon}{2}\right)\frac{1}{|\vec{x}|^{1-\varepsilon}}\,.$
(79)

The second diagram in (76) does not appear at zero temperature (as massless
tadpoles vanish), and can be viewed as a manifestation of the thermal mass. At
order ${\mathcal{O}}(\varepsilon)$, the thermal mass is given by Dolan:1973qd
; Weinberg:1974hy ; Kapusta:2006pm ; Chai:2020zgq

$m_{\text{th}}^{2}=\frac{\varepsilon
N}{2\beta^{2}(N+8)}+\mathcal{O}(\varepsilon^{2})\,,$ (80)

and the diagram is equal to

$\leavevmode\hbox to50.49pt{\vbox
to29.45pt{\pgfpicture\makeatletter\hbox{\hskip 4.76773pt\lower-14.72638pt\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{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{7.8571pt}{2.35703pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-2.35703pt}{14.22638pt}{-4.26773pt}{7.8571pt}{-4.26773pt}{0.0pt}\pgfsys@curveto{-4.26773pt}{-7.8571pt}{-2.35703pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{2.35703pt}{-14.22638pt}{4.26773pt}{-7.8571pt}{4.26773pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{0.0pt}\pgfsys@moveto{45.52458pt}{0.0pt}\pgfsys@curveto{45.52458pt}{1.5715pt}{44.25064pt}{2.84544pt}{42.67914pt}{2.84544pt}\pgfsys@curveto{41.10764pt}{2.84544pt}{39.8337pt}{1.5715pt}{39.8337pt}{0.0pt}\pgfsys@curveto{39.8337pt}{-1.5715pt}{41.10764pt}{-2.84544pt}{42.67914pt}{-2.84544pt}\pgfsys@curveto{44.25064pt}{-2.84544pt}{45.52458pt}{-1.5715pt}{45.52458pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{7.11319pt}\pgfsys@moveto{28.45276pt}{7.11319pt}\pgfsys@curveto{28.45276pt}{11.04173pt}{25.26811pt}{14.22638pt}{21.33957pt}{14.22638pt}\pgfsys@curveto{17.41103pt}{14.22638pt}{14.22638pt}{11.04173pt}{14.22638pt}{7.11319pt}\pgfsys@curveto{14.22638pt}{3.18465pt}{17.41103pt}{0.0pt}{21.33957pt}{0.0pt}\pgfsys@curveto{25.26811pt}{0.0pt}{28.45276pt}{3.18465pt}{28.45276pt}{7.11319pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@moveto{5.69044pt}{0.0pt}\pgfsys@curveto{5.69044pt}{0.78575pt}{5.05348pt}{1.42271pt}{4.26773pt}{1.42271pt}\pgfsys@curveto{3.48198pt}{1.42271pt}{2.84502pt}{0.78575pt}{2.84502pt}{0.0pt}\pgfsys@curveto{2.84502pt}{-0.78575pt}{3.48198pt}{-1.42271pt}{4.26773pt}{-1.42271pt}\pgfsys@curveto{5.05348pt}{-1.42271pt}{5.69044pt}{-0.78575pt}{5.69044pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{0.78575pt}{22.12532pt}{1.42271pt}{21.33957pt}{1.42271pt}\pgfsys@curveto{20.55382pt}{1.42271pt}{19.91685pt}{0.78575pt}{19.91685pt}{0.0pt}\pgfsys@curveto{19.91685pt}{-0.78575pt}{20.55382pt}{-1.42271pt}{21.33957pt}{-1.42271pt}\pgfsys@curveto{22.12532pt}{-1.42271pt}{22.76228pt}{-0.78575pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@fillstroke\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{1}{16}hm_{\text{th}}^{2}\pi^{\frac{\varepsilon-3}{2}}\Gamma\left(-\frac{1+\varepsilon}{2}\right)|\vec{x}|^{1+\varepsilon}\,.$
(81)

The third diagram also appears at zero temperature, and does not receive any
thermal correction

$\leavevmode\hbox to50.49pt{\vbox
to29.45pt{\pgfpicture\makeatletter\hbox{\hskip 4.76773pt\lower-14.72638pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{7.8571pt}{2.35703pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-2.35703pt}{14.22638pt}{-4.26773pt}{7.8571pt}{-4.26773pt}{0.0pt}\pgfsys@curveto{-4.26773pt}{-7.8571pt}{-2.35703pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{2.35703pt}{-14.22638pt}{4.26773pt}{-7.8571pt}{4.26773pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
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}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{4.26773pt}{7.11319pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{4.26773pt}{-7.11319pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@moveto{5.69044pt}{0.0pt}\pgfsys@curveto{5.69044pt}{0.78575pt}{5.05348pt}{1.42271pt}{4.26773pt}{1.42271pt}\pgfsys@curveto{3.48198pt}{1.42271pt}{2.84502pt}{0.78575pt}{2.84502pt}{0.0pt}\pgfsys@curveto{2.84502pt}{-0.78575pt}{3.48198pt}{-1.42271pt}{4.26773pt}{-1.42271pt}\pgfsys@curveto{5.05348pt}{-1.42271pt}{5.69044pt}{-0.78575pt}{5.69044pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{3.69899pt}{7.11319pt}\pgfsys@moveto{5.1217pt}{7.11319pt}\pgfsys@curveto{5.1217pt}{7.89894pt}{4.48474pt}{8.5359pt}{3.69899pt}{8.5359pt}\pgfsys@curveto{2.91324pt}{8.5359pt}{2.27628pt}{7.89894pt}{2.27628pt}{7.11319pt}\pgfsys@curveto{2.27628pt}{6.32744pt}{2.91324pt}{5.69048pt}{3.69899pt}{5.69048pt}\pgfsys@curveto{4.48474pt}{5.69048pt}{5.1217pt}{6.32744pt}{5.1217pt}{7.11319pt}\pgfsys@closepath\pgfsys@moveto{3.69899pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{3.69899pt}{-7.11319pt}\pgfsys@moveto{5.1217pt}{-7.11319pt}\pgfsys@curveto{5.1217pt}{-6.32744pt}{4.48474pt}{-5.69048pt}{3.69899pt}{-5.69048pt}\pgfsys@curveto{2.91324pt}{-5.69048pt}{2.27628pt}{-6.32744pt}{2.27628pt}{-7.11319pt}\pgfsys@curveto{2.27628pt}{-7.89894pt}{2.91324pt}{-8.5359pt}{3.69899pt}{-8.5359pt}\pgfsys@curveto{4.48474pt}{-8.5359pt}{5.1217pt}{-7.89894pt}{5.1217pt}{-7.11319pt}\pgfsys@closepath\pgfsys@moveto{3.69899pt}{-7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.52734375,0.640625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@stroke{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\definecolor[named]{pgffillcolor}{rgb}{0.52734375,0.640625,0.72265625}\pgfsys@color@rgb@fill{0.52734375}{0.640625}{0.72265625}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{0.78575pt}{22.12532pt}{1.42271pt}{21.33957pt}{1.42271pt}\pgfsys@curveto{20.55382pt}{1.42271pt}{19.91685pt}{0.78575pt}{19.91685pt}{0.0pt}\pgfsys@curveto{19.91685pt}{-0.78575pt}{20.55382pt}{-1.42271pt}{21.33957pt}{-1.42271pt}\pgfsys@curveto{22.12532pt}{-1.42271pt}{22.76228pt}{-0.78575pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@fillstroke\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{1}{768}h^{3}\lambda^{\phantom{3}}\pi^{\frac{3(\varepsilon-3)}{2}}\frac{\,\Gamma^{3}\left(\frac{1-\varepsilon}{2}\right)}{\varepsilon(1-3\varepsilon)}\frac{1}{|\vec{x}|^{1-3\varepsilon}}\,.$
(82)

This expression (82) is divergent in the $\varepsilon\to 0$ limit and requires
renormalization. We follow the procedure as explained in Cuomo:2021kfm ;
Gimenez-Grau:2022ebb .

Putting everything together, we obtain

$\displaystyle\frac{\langle\,\phi_{1}(\vec{x}){\mathcal{P}}\,\rangle_{\beta}}{\sqrt{\langle\,\phi_{1}(\infty)\phi_{1}(0)\,\rangle}}=-\frac{\sqrt{N+8}}{2|\vec{x}|}\biggl{[}$