Defects in scalar field theories, RG flows

and Dimensional Disentangling

D. Rodriguez-Gomeza,b<EMAIL_ADDRESS>and J. G. Russo c,d
<EMAIL_ADDRESS>
a Department of Physics, Universidad de Oviedo

C/ Federico García Lorca 18, 33007 Oviedo, Spain

b Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias
(ICTEA)

C/ de la Independencia 13, 33004 Oviedo, Spain.

c Institució Catalana de Recerca i Estudis Avançats (ICREA)

Pg. Lluis Companys, 23, 08010 Barcelona, Spain

d Departament de Física Cuántica i Astrofísica and Institut de Ciències del
Cosmos

Universitat de Barcelona, Martí Franquès, 1, 08028 Barcelona, Spain

ABSTRACT

We consider defect operators in scalar field theories in dimensions
$d=4-\epsilon$ and $d=6-\epsilon$ with self-interactions given by a general
marginal potential. In a double scaling limit, where the bulk couplings go to
zero and the defect couplings go to infinity, the bulk theory becomes
classical and the quantum defect theory can be solved order by order in
perturbation theory. We compute the defect $\beta$ functions to two loops and
study the Renormalization Group flows. The defect fixed points can move and
merge, leading to fixed point annihilation; and they exhibit a remarkable
factorization property where the $\epsilon$-dependence gets disentangled from
the coupling dependence.

###### Contents

  1. 1 Introduction
  2. 2 Defects in $d$-dimensional scalar field theories
  3. 3 Renormalization and $\beta$ functions
    1. 3.1 Relation with the circular defect in $d=4-\epsilon$
  4. 4 One-point function at fixed points
    1. 4.1 General predictions from the Renormalization Group
  5. 5 RG flows in the Twins model
    1. 5.1 Fixed points and RG flows
    2. 5.2 Fixed points to all orders: A Conjecture
  6. 6 Discussion
  7. A Some properties of the one-point function
  8. B Integrals in $d=4-\epsilon$
    1. B.1 The $\mathcal{I}_{1}$ integral
    2. B.2 The $\mathcal{I}_{2}$ integral
  9. C Integrals in $d=6-\epsilon$
    1. C.1 The $\mathcal{I}_{1}$ integral
    2. C.2 The $\mathcal{I}_{2}$ integral

##  1 Introduction

Besides the familiar local operators supported on points, a Quantum Field
Theory can contain extended operators supported on more general manifolds.
These provide the key to the modern notion of generalized symmetry, which is
based on extended operators that depend only topologically on the manifold on
which they are defined. In other instances, defects provide a new window into
the structure of Quantum Field Theory as they can probe aspects of
Renormalization Group (RG) flows or detect new central charges associated to a
given theory. In some physical systems, extended operators can describe
impurities. Thus the study of defects may give important hints on long-
standing questions in condensed matter regarding the ground state, large
distance physics and RG flows in the presence of impurities.

In this paper, we investigate defects in self-interacting scalar field
theories with general classically marginal potentials.333By “classically
marginal” we mean that they are marginal potentials in the classical theory
with $\epsilon=0$. The coupling of the the bulk theory to the extended
operator triggers an RG flow. Our aim is to study this RG flow and the
possible fixed points on which the defect theory may end. A particularly
interesting case is that in which the bulk theory is itself a Conformal Field
Theory (CFT), when the critical defect defines a so-called defect Conformal
Field Theory (dCFT) (see [1] for general properties of dCFT’s). General
properties of line defect RG flows in arbitrary dimensions have been uncovered
in [2], extending the 2d results of [3].

The simplest line defect one may imagine is a trivial line defect (defined by
the insertion of the identity operator on a line, and whose only excitations
are simply the bulk fields restricted to the line), which, as such, is not
very interesting. However, this defect admits exactly one deformation
consisting on a linear combination of the scalars of the theory. Our interest
in this paper is to explore the properties of the theory in the presence of
this line defect. This system has been studied in the past in [4] and revived
more recently in [5, 6, 7].

Stated in full generality, the problem of studying RG flows on line defects in
general scalar field theories (yet with classically marginal potentials) is
very hard. However a particular double-scaling limit of the defect and bulk
couplings was considered in [8], leading to dramatic simplifications. This is
similar in spirit to the large charge limit considered in the literature in
e.g. [9, 10, 11, 12] (a review and further references can be found in [13]),
with direct application to boundary CFT’s in [14]. In the context of Wilson
lines in gauge theories, similar limits have been recently considered in [15,
16, 17, 18, 19, 20, 6, 21]. It is worth noting that the double scaling limit
considered in this paper may not only be a simplifying regime allowing for
detailed studies, as it may also have phenomenological implications. In
particular, as argued in [6] in the case of the $O(3$) model in $d=4-\epsilon$
dimensions, this limit provides an effective description of a large spin
impurity in an antiferromagnet (see also [4, 5]), known as pinning field
defect.

The double-scaling limit has the effect of suppressing the Feynman loop
diagrams involving bulk couplings. Thanks to this dramatic simplification, we
will be able to provide exact formulas for the $\beta$ functions up to two
loops in the defect couplings for theories with arbitrary potential. The RG
flow resulting from these $\beta$ functions turns out to be a gradient flow,
as the $\beta$ functions are the gradient of a scalar function $\mathcal{H}$.

We find a rich structure of fixed points, which merge and disappear though
fixed point annihilation as the ratio of bulk couplings are gradually changed.
Strikingly, to 2 loops order, the $\epsilon$ dependence in the location of the
fixed points factorizes (this means that the $\epsilon$ dependence gets
disentangled –i.e. separated– from the coupling dependence). A close
examination of the higher-order corrections to the $\beta$ functions gives a
strong hint that this property should hold to all orders. This leads to an
exact determination of the fixed points and of the structure of the RG flows
to all loop order.

##  2 Defects in $d$-dimensional scalar field theories

We consider a generic theory with $N$ scalar fields $\varphi_{i}$ in $d$
dimensions with a (classically) marginal potential. The action is

$\mathcal{S}=\int
d^{d}x\left(\frac{1}{2}\big{(}\partial\varphi_{i}\big{)}^{2}+V(\varphi_{i})\right)\,.$
(2.1)

Here $V(\varphi_{i})$ is a degree $n$ polynomial in the scalar fields
$\varphi_{i}$ with couplings denoted generically by $\mathfrak{g}_{\alpha}$,
where $n=4$ in the $d=4-\epsilon$ theory and $n=3$ in the $d=6-\epsilon$
theory.

We now consider, in this theory, a trivial defect supported on a line in the
4d theory and on a two-dimensional plane in the $6d$ theory (that is, denoting
the worldvolume dimension by $d_{\rm wv}$, in 4d, $d_{\rm wv}=1$ and in 6d,
$d_{\rm wv}=2$). To introduce the defect, the theory is deformed by an
operator, which is a linear combination of the scalar fields in the theory.
The deformed action is

$\mathcal{S}\rightarrow\mathcal{S}=\int
d^{d}x\left(\frac{1}{2}\big{(}\partial\varphi_{i}\big{)}^{2}+V(\varphi_{i})+\mathfrak{h}_{i}\,\varphi_{i}\,\delta_{T}(\vec{x})\right)\,,$
(2.2)

where $\delta_{T}(\vec{x})$ is a delta function in the transverse space to the
defect (which is $d-d_{\rm wv}$ dimensional). We will study the RG flow
induced by the defect in a limit where the coefficients $\mathfrak{h}_{i}$ are
large. It is convenient to introduce a parameter $q$ to organize the limit, so
that

$(\mathfrak{g}_{\alpha},\,\mathfrak{h}_{i})=(\frac{g_{\alpha}}{q^{\frac{n-2}{2}}},\,\sqrt{q}\,\nu_{i})\,.$
(2.3)

Upon writing $\varphi_{i}=\sqrt{q}\,\phi_{i}$, we have

$\mathcal{S}=q\,S\,,\qquad S=\int
d^{d}x\left(\frac{1}{2}\big{(}\partial\phi_{i}\big{)}^{2}+V(\phi_{i})+\nu_{i}\,\phi_{i}\,\delta_{T}(\vec{x})\right)\,.$
(2.4)

Let us now consider the limit $q\rightarrow\infty$ with fixed
$g_{\alpha},\,\nu_{i}$. In $d=4-\epsilon$, this corresponds to a double
scaling limit $\mathfrak{h}_{i}\to\infty$, $\mathfrak{g}_{\alpha}\to 0$, with
fixed $\mathfrak{h}_{i}^{2}\mathfrak{g}_{\alpha}$. In $d=6-\epsilon$, the
combination $\mathfrak{h}_{i}\mathfrak{g}_{\alpha}$ is fixed.

In this limit we can use the saddle point method. Clearly, the limit
corresponds to the classical limit of the theory (with $q$ playing the role of
$1/\hbar$) and therefore all bulk loop diagrams will be suppressed (see
section 3 for details). In turn, the defect theory will receive quantum
corrections.

The classical equations of motion are

$\partial^{2}\phi_{i}=\frac{\delta
V}{\delta\phi_{i}}+\nu_{i}\,\delta_{T}(\vec{x})\,.$ (2.5)

In order to do perturbation theory in the $g_{\alpha}$’s, we write
$\phi_{i}=\phi_{i}^{(0)}+\phi_{i}^{(1)}+\phi_{i}^{(2)}\cdots$ where
$\phi_{i}^{(p)}$ is of order $p$ in the $g_{\alpha}$’s. Substituting the
perturbative expansion into the equations of motion and expanding to quadratic
order in the $g_{\alpha}$’s , we obtain

$\partial^{2}\phi_{i}^{(0)}+\partial^{2}\phi_{i}^{(1)}+\partial^{2}\phi_{i}^{(2)}=V_{i}+V_{ij}\phi_{j}^{(1)}+\nu_{i}\,\delta_{T}(\vec{x})\,.$
(2.6)

where we have used the obvious shorthand notation

$V_{i}=\frac{\delta V}{\delta\phi_{i}}\Big{|}_{\phi_{i}^{(0)}}\,,\qquad
V_{ij}=\frac{\delta^{2}V}{\delta\phi_{i}\delta\phi_{j}}\Big{|}_{\phi_{i}^{(0)}}\,.$
(2.7)

Solving order by order, we are led to the equations,

$\partial^{2}\phi_{i}^{(0)}=\nu_{i}\,\delta_{T}(\vec{x})\,,\qquad\partial^{2}\phi_{i}^{(1)}=V_{i}\,,\qquad\partial^{2}\phi_{i}^{(2)}=V_{ij}\,\phi_{j}^{(1)}\,.$
(2.8)

These equations are easily solved by using the Green’s function. One has

$\displaystyle\phi_{i}^{(0)}=-\nu_{i}\int
d^{d}z\,G(x-z)\,\delta_{T}(\vec{z})\,,\qquad\phi_{i}^{(1)}=-\int
d^{d}z\,G(x-z)\,V_{i}(z)\,$ $\displaystyle\phi_{i}^{(2)}=-\int
d^{d}z\,G(x-z)\,V_{ij}(z)\,\phi_{j}^{(1)}(z)\,.$ (2.9)

Let us now evaluate the on-shell action. Using the equations of motion, the
action takes the form

$S_{\rm os}=\int
d^{d}x\left(V-\frac{1}{2}\phi_{i}\,V_{i}+\frac{1}{2}\,\nu_{i}\,\phi_{i}\,\delta_{T}(\vec{x})\right)\,.$
(2.10)

Substituting the perturbative expansion, to quadratic order in the
$g_{\alpha}$’s, we find

$\displaystyle S_{\rm os}$ $\displaystyle=$ $\displaystyle\int
d^{d}x\,\Big{(}-\frac{1}{2}\,\phi^{(0)}_{i}\,V_{i}-\frac{1}{2}\,V_{ij}\,\phi_{i}^{(0)}\,\phi_{j}^{(1)}+\frac{1}{2}\,V_{i}\,\phi_{i}^{(1)}+V$
(2.11) $\displaystyle+$
$\displaystyle\frac{1}{2}\,\nu_{i}\,\phi^{(0)}_{i}\,\delta_{T}(\vec{x})+\frac{1}{2}\,\nu_{i}\,\phi^{(1)}_{i}\,\delta_{T}(\vec{x})+\frac{1}{2}\,\nu_{i}\,\phi^{(2)}_{i}\,\delta_{T}(\vec{x})\Big{)}\,.$

From the explicit form of the perturbative solution one can easily show that

$\int
d^{d}x\,\frac{1}{2}\,\nu_{i}\,\phi^{(1)}_{i}(x)\,\delta_{T}(\vec{x})=\int
d^{d}x\,\frac{1}{2}\,V_{i}(x)\,\phi_{i}^{(0)}(x)\,,$ (2.12)

and

$\int
d^{d}x\,\frac{1}{2}\,\nu_{i}\,\phi^{(2)}_{i}(x)\,\delta_{T}(\vec{x})=\int
d^{d}x\,\frac{1}{2}\,V_{ij}(x)\,\phi_{i}^{(0)}(x)\,\phi_{j}^{(1)}(x)\,.$
(2.13)

Hence

$S_{\rm os}=\int
d^{d}x\,\left(V+\frac{1}{2}V_{i}\,\phi_{i}^{(1)}+\frac{1}{2}\,\nu_{i}\,\phi^{(0)}_{i}\,\delta_{T}(\vec{x})\right)\,.$
(2.14)

In order to further proceed, we note that
$\phi_{i}^{(0)}=\frac{\nu_{i}}{\nu_{1}}\,\phi_{1}^{(0)}$. Moreover, since $V$,
$V_{i}$ are homogeneous functions of degree $n$ and $n-1$ respectively, it
follows that

$V=V(\phi_{i}^{(0)})=V(\nu_{i})\,\frac{(\phi_{1}^{(0)})^{n}}{\nu_{1}^{n}}\,,\qquad
V_{i}=\frac{\delta
V}{\delta\phi_{i}}(\phi_{i}^{(0)})=V_{i}(\nu_{i})\,\frac{(\phi_{1}^{(0)})^{n-1}}{\nu_{1}^{n-1}}\,.$
(2.15)

Therefore

$\phi_{i}^{(1)}=-\frac{V_{i}(\nu_{i})}{\nu_{1}^{n-1}}\,\int
d^{d}y\,G(x-y)\,(\phi_{1}^{(0)})^{n-1}(y)\,.$ (2.16)

Then

$\displaystyle S_{\rm os}$ $\displaystyle=$
$\displaystyle\frac{V}{\nu_{1}^{n}}\int
d^{d}x\,(\phi_{1}^{(0)})^{n}(x)-\frac{V_{i}^{2}}{2\nu_{1}^{2(n-1)}}\,\int
d^{d}xd^{d}y\,G(x-y)\,(\phi_{1}^{(0)})^{n-1}(x)(\phi_{1}^{(0)})^{n-1}(y)$
(2.17) $\displaystyle+$ $\displaystyle\frac{1}{2}\,\nu_{i}\,\int
d^{d}x\,\phi^{(0)}_{i}\,\delta_{T}(\vec{x})\,,$

where $V_{i}^{2}=\sum_{i}V_{i}V_{i}$ and in $V$, $V_{i}$ the $\phi_{i}$ are
replaced by $\nu_{i}$. Using now the explicit form of $\phi_{i}^{(0)}$, we get

$S_{\rm
os}=-\frac{1}{2}\,\nu_{i}^{2}\,I_{0}+(-1)^{n}\,V(\nu_{i})\,I_{1}-\frac{V_{i}^{2}}{2}\,I_{2}\,,$
(2.18)

where

$\displaystyle I_{0}=\int d^{d}z_{1}\int
d^{d}z_{2}\,G(z_{2}-z_{2})\,\delta_{T}(\vec{z}_{1})\delta_{T}(\vec{z}_{2})\,,$
(2.19) $\displaystyle I_{1}=\int d^{d}x\,\int d^{d}z_{1}\cdots\int
d^{d}z_{n}\,G(x-z_{1})\cdots
G(x-z_{n})\,\delta_{T}(\vec{z}_{1})\cdots\delta_{T}(\vec{z}_{n})\,,$ (2.20)
$\displaystyle I_{2}=\int d^{d}x\int d^{d}y\int d^{d}u_{1}\cdots\int
d^{d}u_{n-1}\int d^{d}v_{1}\cdots\int d^{d}v_{n-1}G(x-y)$ (2.21)
$\displaystyle G(x-u_{1})\cdots G(x-u_{n-1})\,G(y-v_{1})\cdots
G(y-v_{n-1})\,\delta_{T}(\vec{u}_{1})\cdots\delta_{T}(\vec{u}_{n-1})\,\delta_{T}(\vec{v}_{1})\cdots\delta_{T}(\vec{v}_{n-1})\,.$

The factor $(-1)^{n}$ in the second term in (2.18) would give a $(-1)$ factor
in the $d=6-\epsilon$ theory. However, since in this case the potential is
odd, we can redefine the scalar fields $\phi_{i}\to-\phi_{i}$ and set this
factor to $+1$, so in what follows this factor will be removed.

Fourier-transforming one has

$I_{0}=\mathcal{V}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\frac{1}{|\vec{p}_{T}|^{2}}\,,\qquad
I_{1}=\mathcal{V}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\frac{1}{|\vec{p}_{T}|^{2}}\,\mathcal{I}_{1}\,,\qquad
I_{2}=\mathcal{V}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\frac{1}{|\vec{p}_{T}|^{2}}\,\mathcal{I}_{2}\,,$
(2.22)

where $\mathcal{V}$ is the regularized volume of the defect worldvolume,
$d_{T}=d-d_{\rm wv}$ and $d\vec{p}_{T}=d^{d_{T}}p$, represents the volume
element in momentum space transverse to the defect and

$\mathcal{I}_{1}=\int\frac{d\vec{k}_{T}^{1}}{(2\pi)^{d_{T}}}\cdots\int\frac{d\vec{k}_{T}^{n-2}}{(2\pi)^{d_{T}}}\frac{1}{(\vec{k}_{T}^{1})^{2}\cdots(\vec{k}^{n-2}_{T})^{2}\,(\vec{p}_{T}-\vec{k}_{T}^{1}-\cdots-\vec{k}_{T}^{n-2})^{2}}\,,$
(2.23)

and

$\displaystyle\mathcal{I}_{2}=\int\frac{d\vec{k}_{T}^{1}}{(2\pi)^{d_{T}}}\cdots\int\frac{d\vec{k}_{T}^{n-1}}{(2\pi)^{d_{T}}}\int\frac{d\vec{q}_{T}^{\
1}}{(2\pi)^{d_{T}}}\cdots\int\frac{d\vec{q}_{T}^{\
n-3}}{(2\pi)^{d_{T}}}\frac{1}{(\vec{k}_{T}^{1})^{2}\cdots\,(\vec{k}_{T}^{n-1})^{2}}\,$
(2.24)
$\displaystyle\frac{1}{(\vec{k}_{T}^{n-1}-\vec{k}_{T}^{1}-\cdots-\vec{k}_{T}^{n-2})^{2}\,(\vec{q}_{T}^{\
1})^{2}\cdots(\vec{q}_{T}^{\ n-3})^{2}(\vec{k}_{T}^{n-1}+\vec{q}^{\
1}_{T}+\cdots+\vec{q}^{\ n-3}_{T}-\vec{p}_{T})^{2}}\,.$

The integrals (2.23) and (2.24) are computed in the appendix. We obtain

$\mathcal{I}_{1}=\frac{\Omega}{\epsilon}-2\,c\,\Omega\,\log|\vec{p}_{T}|+\epsilon\,A+\epsilon\,2\,c^{2}\,\Omega\,(\log|\vec{p}_{T}|)^{2}+O(\epsilon^{2})\,,$
(2.25)

and

$\mathcal{I}_{2}=\frac{\Omega^{2}}{2\epsilon^{2}}+\frac{c\,\Omega^{2}\,(c-2\,\log|\vec{p}_{T}|)}{\epsilon}+\Big{(}C-4\,c^{3}\,\Omega^{2}\,\log|\vec{p}_{T}|+4\,c^{2}\,\Omega^{2}\,(\log|\vec{p}_{T}|)^{2}\Big{)}+O(\epsilon)\,,$
(2.26)

where $A,\,C$ are (scheme-dependent) numerical constants. The table (2.27)
summarizes the values of the different constants in the four-dimensional and
six-dimensional theories.

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr d&\Omega&n&c&A&C\\\ \hline\cr
4&\frac{1}{32\pi^{2}}&4&1&\frac{9\,\Omega}{2}-\frac{7}{768}&\frac{19\,\Omega^{2}}{2}-\frac{11\,\Omega}{768}\\\
\hline\cr
6&\frac{1}{8\pi^{2}}&3&\frac{1}{2}&\frac{\Omega}{2}-\frac{1}{384}&\frac{7\,\Omega^{2}}{8}-\frac{\Omega}{384}\\\
\hline\cr\end{array}$ (2.27)

Note that in $\mathcal{I}_{1}$ we have kept a term of order $\epsilon$, whose
relevance will become clear below.

Coming back to our computation, the on-shell action takes the form, $S_{\rm
os}={\cal V}\,s_{\rm os}$, with

$\displaystyle s_{\rm os}$ $\displaystyle=$
$\displaystyle\Big{(}-\frac{\nu_{i}^{2}}{2}+\frac{2\,\Omega\,V-c^{2}\,\Omega^{2}\,V_{i}^{2}}{2\,\epsilon}-\frac{\Omega^{2}\,V_{i}^{2}}{4\,\epsilon^{2}}\Big{)}\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\frac{1}{|\vec{p}_{T}|^{2}}$
$\displaystyle-\Big{(}2\,c\,\Omega\,V-\frac{c\,\Omega^{2}\,V_{i}^{2}}{\epsilon}\Big{)}\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\frac{\log|\vec{p}_{T}|}{|\vec{p}_{T}|^{2}}$
$\displaystyle-\frac{V_{i}^{2}}{2}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,\Big{(}C-4\,c^{3}\,\Omega^{2}\,\log|\vec{p}_{T}|+4\,c^{2}\,\Omega^{2}\,(\log|\vec{p}_{T}|)^{2}\Big{)}$
$\displaystyle+V\,\epsilon\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{p_{T}^{2}}\,\Big{(}A+2\,c^{2}\,\Omega\,(\log|\vec{p}_{T}|)^{2}\Big{)}\,.$

The factors of $\Omega$ in (2) could be removed by a rescaling $\Omega
g_{\alpha}\to g_{\alpha}$ along with a redefinition of the numerical constants
$C$, $A$. However, we find more convenient to keep the explicit dependence on
$\Omega$ in all formulas.

##  3 Renormalization and $\beta$ functions

The defect breaks translational invariance in the transverse directions. As a
consequence, one-point functions in the bulk can be non-vanishing.444 For
critical defects in bulk conformal theories, the $SO(d_{\rm wv}+1,1)\times
SO(d-d_{\rm wv})$ conformal group does not imply the vanishing of the one-
point function but it places other constraints on the structure of correlation
functions. See e.g. [1] for a detailed discussion. The integrated one-point
function of the scalar in the presence of the defect can be computed by
differentiating with respect to the defect coupling as

$\frac{1}{Z}\,\frac{\delta Z}{\delta\mathfrak{h}_{i}}=-\langle\int
d^{d}x\,\varphi_{i}\,\delta_{T}(\vec{x})\rangle\qquad\longrightarrow\qquad\frac{1}{Z}\,\frac{\delta
Z}{\delta\nu_{i}}=-q\,\langle\int
d^{d}x\,\phi_{i}\,\delta_{T}(\vec{x})\rangle\,.$ (3.1)

In appendix A we review some basic facts about one-point functions in defect
theories. Related calculations of the 1-point function can be found in [4, 5,
6, 7]. In the large $q$ limit, this VEV is computed by

$\langle\int d^{d}x\,\phi_{i}\,\delta_{T}(\vec{x})\rangle=-{\cal
V}\,\partial_{i}s_{\rm os}\,,$ (3.2)

where $\partial_{i}$ is the derivative with respect to $\nu_{i}$. Using (2),
we obtain

$\displaystyle\partial_{i}s_{\rm os}$ $\displaystyle=$
$\displaystyle\Big{(}-\nu_{i}+\frac{\Omega\,V_{i}-c^{2}\,\Omega^{2}\,V_{j}\,V_{ij}}{\epsilon}-\frac{\Omega^{2}\,V_{j}\,V_{ij}}{2\,\epsilon^{2}}\Big{)}\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\frac{1}{|\vec{p}_{T}|^{2}}$
$\displaystyle-2\,c\,\Big{(}\Omega\,V_{i}-\frac{\Omega^{2}\,V_{j}\,V_{ij}}{\epsilon}\Big{)}\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\frac{\log|\vec{p}_{T}|}{|\vec{p}_{T}|^{2}}$
$\displaystyle-
V_{j}\,V_{ij}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,\Big{(}C-4\,c^{3}\,\Omega^{2}\,\log|\vec{p}_{T}|+4\,c^{2}\,\Omega^{2}\,(\log|\vec{p}_{T}|)^{2}\Big{)}$
$\displaystyle+V_{i}\,\epsilon\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{p_{T}^{2}}\,\Big{(}A+2\,c^{2}\,\Omega\,(\log|\vec{p}_{T}|)^{2}\Big{)}\,.$

This represents the Fourier-transformed one-point function in the bulk (see
appendix A for further discussions).

The content of (3) as well as the simplifications that occur in this limit can
be understood diagrammatically from fig. 1, where we list all diagrams
contributing to the one-point function. The vertices can be read from (2.18),
by computing $\partial_{i}S_{\rm os}$. The second and third diagrams in the
first line (green background) give contributions proportional to the integrals
$I_{1}$ and $I_{2}$, respectively. The diagrams in the second line in fig. 1
(in red background) are $\mathcal{O}(g_{\alpha}g_{\beta})$ terms which are not
contributing to (3). They represent loops in the bulk couplings $g_{\alpha}$,
which are suppressed in the large $q$ (or classical bulk) limit considered
here.555The explicit $q$-dependence is as follows. The diagrams in green are
proportional to
$\mathfrak{h}_{i}\,(\mathfrak{h_{j}}\mathfrak{h_{k}}\mathfrak{g}_{\alpha})^{n}\sim
q^{+\frac{1}{2}}$, $n=0,1,2$, while those in red are proportional to
$\mathfrak{h}_{i}\mathfrak{g}_{\alpha}\mathfrak{g}_{\beta}\sim
q^{-\frac{3}{2}}$,
$\mathfrak{h}_{i}\mathfrak{h}_{j}\mathfrak{h}_{k}\mathfrak{g}_{\alpha}\mathfrak{g}_{\beta}\sim
q^{-\frac{1}{2}}$.

Figure 1: Diagrams contributing to the one-point function of the scalar.
Squares correspond to the bulk coupling; circles to defect couplings. In the
double scaling limit only the diagrams with green background contribute, while
those in red are suppressed.

We obtained (3) from the derivative of the defect partition function. This
derivative is “cut-opening” the vacuum diagrams. A technical remark is that,
in principle, to $\mathcal{O}(g_{\alpha}g_{\beta})$, this “cut-opening” of
diagrams could be done in two ways, as shown in fig. 2 below. These correspond
to the two possible ways of extracting an overall
$\frac{1}{|\vec{p}_{T}|^{2}}$ when writing the Fourier-transform in (2.22).
The way a) corresponds to the expression of the integrals $I_{2}$, with ${\cal
I}_{2}$ given by (2.24). The way b) does not represent a contribution to the
one-point function.

Figure 2: Diagrams contributing to $\partial_{i}s_{\rm os}$ can be obtained by
cutting one line of the vacuum diagram of the defect theory. In this example,
there are two possible cuts, but only (a) represents a contribution to the
one-point function.

Let us now proceed to the renormalization of the UV divergences by demanding
the one-point function to be finite. To get rid of the divergences in (3) we
introduce the renormalized coupling

$\nu_{iR}=\nu_{i}-\frac{\Omega\,V_{i}-c^{2}\,\Omega^{2}\,V_{j}\,V_{ij}}{\epsilon}+\frac{\Omega^{2}\,V_{j}\,V_{ij}}{2\,\epsilon^{2}}\,.$
(3.4)

Inverting this equation to quadratic order in the $g_{\alpha}$’s, one finds

$\nu_{i}=\nu_{iR}+\frac{\Omega\,V^{R}_{i}-c^{2}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}}{\epsilon}+\frac{\Omega^{2}\,V^{R}_{ij}\,V^{R}_{j}}{2\,\epsilon^{2}}\,,$
(3.5)

where the superscript $R$ stands for evaluation in $\nu_{iR}$.

Having defined the renormalized coupling $\nu_{iR}$, the coefficient of
$1/|\vec{p}_{T}|^{2}$ is rendered finite. It should be noted that there are
still potential divergences in (3) and no other parameter to take care of
them. However, by writing $\nu_{i}$ in terms of the renormalized coupling one
finds the relation

$\Omega\,V_{i}-\frac{\Omega^{2}\,V_{j}\,V_{ij}}{\epsilon}=\Omega\,V^{R}_{i}\,,$
(3.6)

Thus, the coefficient of $(\log|\vec{p}_{T}|)/|\vec{p}_{T}|^{2}$ is made
finite by the same renormalization prescription.

Let us now compute the $\beta$ functions. Restoring the renormalization scale,
we have

$\nu_{i}=\mu^{\frac{\epsilon}{2}}\,(\nu_{iR}+F_{i})\,,\qquad
F_{i}\equiv\frac{1}{\epsilon}\big{(}\Omega\,V^{R}_{i}-c^{2}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}\big{)}+\frac{1}{2\,\epsilon^{2}}\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}\,.$
(3.7)

Using that the bare couplings $\nu_{i}$ are independent of $\mu$, we obtain
the relation

$\mu\frac{d\nu_{i}}{d\mu}=0\quad\implies\quad\frac{\epsilon}{2}\,\nu_{iR}+\frac{\epsilon}{2}\,F_{i}+\beta_{i}+\frac{\partial
F_{i}}{\partial\nu_{jR}}\,\beta_{j}+\frac{\partial F_{i}}{\partial
g_{\alpha}}\,\beta_{g_{\alpha}}=0\,.$ (3.8)

Because bulk loops are suppressed, the bulk couplings $g_{\alpha}$ are not
renormalized and the corresponding $\beta_{g_{\alpha}}$ ’s are just given by
their classical expressions $\beta_{g_{\alpha}}=-c\,\epsilon\,g_{\alpha}$,
determined by dimensional analysis. Using also the fact that the $p$-th
derivative $V^{R}_{i_{1}\cdots i_{p}}$ is linear in the $g_{\alpha}$’s, it
follows that

$\frac{\partial F_{i}}{\partial
g_{\alpha}}\,\beta_{g_{\alpha}}=-c\,\epsilon\,F_{i}+c\,\Omega^{2}\,V_{j}^{R}\,V_{ij}^{R}\,\Big{(}c^{2}-\frac{1}{2\,\epsilon}\Big{)}\,.$
(3.9)

Thus

$\frac{\epsilon}{2}\,\nu_{iR}+\beta_{i}+\frac{\partial
F_{i}}{\partial\nu_{jR}}\,\beta_{j}-\epsilon\,\Big{(}c-\frac{1}{2}\Big{)}\,F_{i}+c\,\Omega^{2}\,V_{j}^{R}\,V_{ij}^{R}\,\Big{(}c^{2}-\frac{1}{2\,\epsilon}\Big{)}=0\,.$
(3.10)

On dimensional grounds, one has

$\beta_{i}=-\frac{\epsilon}{2}\nu_{iR}+\beta_{i}^{(Q)}\,,$ (3.11)

where $\beta_{i}^{(Q)}$ represents the quantum contributions. Substituting
this expression into (3.10), we find the following equation for
$\beta_{i}^{(Q)}$

$\beta_{i}^{(Q)}+\frac{\partial
F_{i}}{\partial\nu_{jR}}\,\beta_{j}^{(Q)}=\epsilon\,\Big{(}c-\frac{1}{2}\Big{)}\,F_{i}+\frac{\epsilon}{2}\nu_{jR}\frac{\partial
F_{i}}{\partial\nu_{jR}}-c\,\Omega^{2}\,V_{j}^{R}\,V_{ij}^{R}\,\Big{(}c^{2}-\frac{1}{2\,\epsilon}\Big{)}\,.$
(3.12)

The $p$-th derivative of $V^{R}$ is a homogeneous polynomial of degree $n-p$
in $\nu_{iR}$. Therefore

$\nu_{jR}\frac{\partial V^{R}_{i_{1}\cdots
i_{p}}}{\partial\nu_{jR}}=(n-p)V^{R}_{i_{1}\cdots i_{p}}\,,$ (3.13)

and

$\nu_{jR}\frac{\partial
F_{i}}{\nu_{jR}}=(n-1)\,F_{i}-(n-2)\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}\,\Big{(}\frac{c^{2}}{\epsilon}-\frac{1}{2\,\epsilon^{2}}\Big{)}\,.$
(3.14)

Thus

$\beta_{i}^{(Q)}+\frac{\partial
F_{i}}{\partial\nu_{jR}}\,\beta_{j}^{(Q)}=2\,c\,\Omega\,V^{R}_{i}-4c^{3}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}+\frac{2\,c}{\epsilon}\,\Omega^{2}V^{R}_{j}\,V^{R}_{ij}\,.$
(3.15)

Using the definition of $F_{i}$ in (3.7), we get

$\frac{\partial
F_{i}}{\partial\nu_{jR}}\,\beta_{j}^{(Q)}=\frac{\Omega\,V_{ij}^{R}}{\epsilon}\,\beta_{j}^{(Q)}+...\,,$
(3.16)

where we neglected $O(g^{3})$ terms (recall that $\beta_{j}^{(Q)}$ is of order
$g$, and every factor $V_{i...}^{R}$ adds a power of $g$). Therefore

$\beta_{i}^{(Q)}+\frac{\Omega\,V_{ij}^{R}}{\epsilon}\,\beta_{j}^{(Q)}=2\,c\,\Omega\,V^{R}_{i}-4c^{3}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}+\frac{2\,c}{\epsilon}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}\,.$
(3.17)

The solution to this equation is

$\beta_{i}^{(Q)}=2\,c\,\Omega\,V^{R}_{i}-4c^{3}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}\,.$
(3.18)

Thus, restoring the classical piece, the $\beta$ functions are

$\beta_{i}=-\frac{\epsilon}{2}\nu_{iR}+2\,c\,\Omega\,V^{R}_{i}-4c^{3}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}\,.$
(3.19)

We note that the $\beta$ functions can be written as
($\vec{\partial}=\frac{\vec{\partial}}{\partial\vec{\nu}_{R}}$)

$\vec{\beta}=2\,c\,\vec{\partial}\mathcal{H}\,,\qquad\mathcal{H}=-\frac{\epsilon}{8\,c}\nu_{iR}^{2}+\Omega\,V^{R}-c^{2}\,\Omega^{2}\,(V^{R}_{i})^{2}\,.$
(3.20)

As a check, the $\beta$ functions in (3.20) reproduce the results in [4, 5, 8]
when particularized to the corresponding models.

Eq.(3.20) shows that the RG flow is a gradient flow of the ”energy function”
$\mathcal{H}$ in both in $d=4-\epsilon$ and in $d=6-\epsilon$. Interestingly,
the effect of the classical contribution to the $\beta$ functions is to
generate a negative mass squared contribution to the “energy” $\mathcal{H}$.
If the potential $V^{R}$ (and consequently $(V_{i}^{R})^{2}$) has a global
symmetry, then the theory at the fixed points will exhibit a symmetry breaking
pattern similar to that of the Higgs mechanism. For instance, if the theory
has an $O(N)$ symmetry, then, at non-trivial fixed points only an $O(N-1)$
subgroup will be unbroken.666This property was implicitly used in e.g. [4, 5,
8].

###  3.1 Relation with the circular defect in $d=4-\epsilon$

The previous computation can be adapted to circular defects of radius $R$ in
$d=4-\epsilon$. The on-shell action is

$S_{\rm
os}^{\circ}=-\frac{1}{2}\,\nu_{i}^{2}\,I_{0}+V\,I_{1}-\frac{V_{i}^{2}}{2}\,I_{2}\,,$
(3.21)

where now the integrals are to be computed with a $\delta_{T}$ function
localizing not to a line defect but to the circular defect. To one-loop order,
these integrals can be borrowed from [5, 7]. Explicitly,

$I_{0}=-\frac{\epsilon}{4}\,,\qquad I_{1}=-2\,\Omega\,.$ (3.22)

Writing (3.21) in terms of the renormalized coupling and using (3.13), for
$n=4$, $c=1$ we find the remarkable formula

$S_{\rm os}^{\circ}=-\mathcal{H}\,.$ (3.23)

Thus $\exp(\mathcal{H})$ represents the VEV of the circular defect. The
formula $\vec{\beta}=2\vec{\partial}\mathcal{H}=-2\vec{\partial}S_{\rm
os}^{\circ}$ agrees with general arguments given in [2, 5, 15], which applies
to line defects. For the surface defect in 6d, we found the gradient formula
(3.20), but the analogous interpretation of $\mathcal{H}$ as the on-shell
action of a compact defect remains to be understood (for additional comments,
see section 6).

##  4 One-point function at fixed points

Having renormalized the UV divergences, we can now write (3) in terms of the
renormalized couplings:

$\displaystyle\partial_{i}s_{\rm os}$ $\displaystyle=$
$\displaystyle-\nu_{iR}\,\Big{[}\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\frac{1}{|\vec{p}_{T}|^{2}}\Big{(}1+\frac{(C-\Omega\,A)\,V_{j}^{R}\,V_{ij}^{R}}{\nu_{iR}}-\frac{V_{i}^{R}\,\epsilon\,A}{\nu_{iR}}\Big{)}$
$\displaystyle+\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,\Big{(}\frac{2\,c\,\Omega\,V^{R}_{i}-4\,c^{3}\,\Omega^{2}\,V_{j}^{R}\,V_{ij}^{R}}{\nu_{iR}}\Big{)}\,\log|\vec{p}_{T}|$
$\displaystyle+\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,\Big{(}\frac{2\,c^{2}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}}{\nu_{iR}}-\frac{2\,c^{2}\,\Omega\,V_{i}^{R}\,\epsilon}{\nu_{iR}}\Big{)}(\log|\vec{p}_{T}|)^{2}\Big{]}\,.$

Here we see the relevance of keeping the $O(\epsilon)$ term in the integral
(2.25): it contributes to the finite coefficient in the
$V^{R}_{j}\,V^{R}_{ij}$ term multiplying $(\log|\vec{p}_{T}|)^{2}$. The
$\epsilon$ factor canceled out against a $1/\epsilon$ factor coming from a
term in the formula (3.5) for $\nu_{i}$.

We shall also keep terms proportional to $\epsilon$ in (4), as they will be
important when evaluating this expression at the defect fixed point, that is,
at the solution of

$\beta_{i}=-\frac{\epsilon}{2}\nu_{iR}+2\,c\
\Omega\,V^{R}_{i}-4c^{3}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}=0\,.$ (4.2)

There is a fixed point at $\nu_{iR}=0$ and non-trivial fixed points where
$g\,(\nu^{R})^{n-2}$ are of order $\epsilon$. Here the order $\epsilon$ terms
in (4) are also important, since, at the fixed point, they will contribute
just like the $V^{R}_{j}\,V^{R}_{ij}$ terms, giving rise to $O(\epsilon^{2})$
contributions.777It is easy to check that terms with higher powers of
$\epsilon$ in the integrals (2.25) and (2.26) do not contribute to the order
$O(\epsilon^{2})$ studied here.

Let us now compute (4) at a non-trivial fixed point. The fact that
$g\,(\nu^{R})^{n-2}\sim\epsilon$ implies that the factor multiplying
$\frac{1}{|\vec{p}_{T}|^{2}}$ in the first line of (4) is of the form
$1+\epsilon^{2}$. Hence, to order $\epsilon^{2}$ we can write

$\displaystyle\partial_{i}s_{\rm os}$ $\displaystyle=$
$\displaystyle-\nu_{iR}\,\Big{(}1+\frac{(C-\Omega\,A)\,V_{j}^{R}\,V_{ij}^{R}}{\nu_{iR}}-\frac{V_{i}^{R}\,\epsilon\,A}{\nu_{iR}}\Big{)}$
$\displaystyle\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,\Big{[}1+\Big{(}\frac{2\,c\,\Omega\,V^{R}_{i}-4\,c^{3}\,\Omega^{2}\,V_{j}^{R}\,V_{ij}^{R}}{\nu_{iR}}\Big{)}\,\log|\vec{p}_{T}|$
$\displaystyle+\Big{(}\frac{2\,c^{2}\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}}{\nu_{iR}}-\frac{2\,c^{2}\,\Omega\,V_{i}^{R}\,\epsilon}{\nu_{iR}}\Big{)}(\log|\vec{p}_{T}|)^{2}\Big{]}\,.$

Squaring the $\beta$ function equation (4.2) and taking its derivative, to
quadratic order in the $g_{\alpha}$’s (or, equivalently, to order
$\epsilon^{2}$), one finds the relation

$2\beta_{j}\,\partial_{i}\beta_{j}+n\,\epsilon\,\beta_{i}=(1-n)\,\frac{\epsilon^{2}}{2}\nu_{iR}+8\,c^{2}\,(1+c^{2}\,\epsilon)\,\Omega^{2}\,V^{R}_{j}\,V^{R}_{ij}\,.$
(4.4)

Thus, at the fixed point, and to order $\epsilon^{2}$, we finally obtain

$\partial_{i}s_{\rm
os}=-\nu_{iR}\,\mathcal{N}\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,\Big{[}1+\frac{\epsilon}{2}\,\log|\vec{p}_{T}|+\frac{1}{8}(1-2c)\,\epsilon^{2}\,(\log|\vec{p}_{T}|)^{2}\Big{]}\,,$
(4.5)

with

$\mathcal{N}=\Big{(}1+\frac{(C-\Omega\,A)\,V_{j}^{R}\,V_{ij}^{R}}{\nu_{iR}}-\frac{V_{i}^{R}\,\epsilon\,A}{\nu_{iR}}\Big{)}=1+\frac{(1+2c)\,C-(1+6c)\,\Omega\,A}{16\,c^{2}\,\Omega^{2}}\,\epsilon^{2}\,.$
(4.6)

In the last equality we used that we are sitting at a non-trivial fixed point.
The constant $\mathcal{N}$ is scheme dependent.

The one-point function (4.5) shows a deviation from the conformal behavior at
order $\mathcal{O}(\epsilon^{2})$. One formally recovers the CFT behavior by
setting $c\to 0$. This is consistent with the fact that $c$ appears as a
coefficient in the bulk coupling $\beta$ function. Recall that in the double-
scaling limit studied here the running of bulk couplings is completely
determined by the classical contribution to the $\beta$ functions, dictated by
dimensional analysis.

###  4.1 General predictions from the Renormalization Group

It is possible to understand the formula (4.5) from a deeper perspective. Let
us consider the quantity $\hat{G}_{i}\equiv\partial_{i}s_{\rm os}$. Explicitly

$\hat{G}_{i}=-\mathcal{V}^{-1}\,\int
d^{d}x\,\langle\phi\rangle\,\delta_{T}(\vec{x})\,.$ (4.7)

Since $\hat{G}_{i}$ has dimension $d_{\rm wv}-\frac{\epsilon}{2}$ and must
vanish when $\nu_{iR}=0$, it can be written as

$\hat{G}_{i}=-\nu_{iR}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,\hat{F}(\frac{|\vec{p}_{T}|}{\mu},\mu^{n-c\,d}\,g_{\alpha},\,\mu^{-\frac{\epsilon}{2}}\,\nu_{iR})\,,$
(4.8)

where $\hat{F}$ is a dimensionless function of its arguments.

Scalar fields do not get anomalous dimension because loops of the bulk
coupling are suppressed in the double-scaling limit. Then, the function
$\hat{G}_{i}$ satisfies the Callan-Symanzik equation

$\mu\,\frac{d\hat{G}_{i}}{d\mu}=-\frac{\epsilon}{2}\,\hat{G}_{i}\,.$ (4.9)

Since $\hat{G}_{i}$ is a function of
$\\{\mu,\,g_{\alpha}(\mu),\,\nu_{iR}(\mu)\\}$, this equation becomes

$\mu\frac{\partial\hat{G}_{i}}{\partial\mu}+\frac{\epsilon}{2}\,\hat{G}_{i}+\beta_{g_{\alpha}}\,\frac{\partial\hat{G}_{i}}{\partial
g_{\alpha}}+\beta_{j}\,\frac{\partial\hat{G}_{i}}{\partial\nu_{jR}}=0\,.$
(4.10)

We can cross-check these equations, in particular, the RHS in (4.9), in the
case of a dCFT, where $\beta_{g_{\alpha}}=\beta_{i}=0$. Then, the Callan-
Symanzik equation (4.10) gives
$\hat{F}=(|\vec{p}_{T}|/\mu)^{\frac{\epsilon}{2}}$, leading to

$\hat{G}_{i}=-\nu_{iR}\,\mu^{-\frac{\epsilon}{2}}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2-\frac{\epsilon}{2}}}\,.$
(4.11)

This can be written as

$\hat{G}_{i}=-\mathcal{V}^{-1}\,\int
dx\,\langle\phi\rangle\,\delta_{T}(\vec{x})\,,\qquad\langle\phi\rangle=\nu_{iR}\,\mu^{-\frac{\epsilon}{2}}\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{e^{i\vec{p}_{T}\cdot\vec{x}_{T}}}{|\vec{p}_{T}|^{2-\frac{\epsilon}{2}}}$
(4.12)

which is the expected form for the 1-point function (compare with e.g. (21) in
[4] in the case of vanishing anomalous dimension).

Coming back to our case, in perturbation theory

$\hat{F}=\sum_{k=0}^{\infty}\,(\mu^{n-c\,d}\,g_{\alpha})^{k}\,\hat{f}_{k}(\frac{|\vec{p}_{T}|}{\mu},\,\mu^{-\frac{\epsilon}{2}}\,\nu_{iR})\,.$
(4.13)

Here $(g_{\alpha})^{k}$ is a symbolic notation for terms involving $k$ of the
$g$’s, such as $g_{\alpha_{1}}...g_{\alpha_{k}}$.888 A more proper notation
would be to write
$\hat{F}=\sum_{k=0}^{\infty}\sum_{\alpha_{1},\cdots\alpha_{k}}(\mu^{n-c\,d}\,g_{\alpha_{1}})\cdots(\mu^{n-c\,d}\,g_{\alpha_{k}})\hat{f}_{k}^{\alpha_{1}\cdots\alpha_{k}}$.
We choose to condense the notation in the obvious way in order not to clutter
the presentation. Then

$g_{\alpha}\frac{\partial\hat{f}}{\partial
g_{\alpha}}=\sum_{k=1}^{\infty}\,k\,\hat{F}_{k}\,,\
\qquad\hat{F}_{k}\equiv(\mu^{n-c\,d}\,g_{\alpha})^{k}\,\hat{f}_{k}\ .$ (4.14)

Using that $\beta_{g_{\alpha}}=-c\,\epsilon\,g_{\alpha}$, we get

$\beta_{g_{\alpha}}\frac{\partial\hat{G}_{i}}{\partial
g_{\alpha}}=-\nu_{iR}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,\sum_{k=1}^{\infty}\,(-c\,\epsilon)\,k\,\hat{F}_{k}\,.$
(4.15)

Thus, at the defect fixed point, the Callan-Symanzik equation (4.10) becomes

$\sum_{k=0}^{\infty}\Big{(}\mu\frac{\partial\hat{F}_{k}}{\partial\mu}+\frac{\epsilon}{2}\hat{F}_{k}\Big{)}-c\,\epsilon\,\sum_{k=1}^{\infty}\,k\,\hat{F}_{k}=0\,.$
(4.16)

At the defect fixed point, the $\nu_{iR}$ are solutions of $\beta_{i}=0$,
given in (4.2). In the $\epsilon$ expansion, the solution is of the form
$g_{\alpha}\nu_{i}\nu_{k}=O(\epsilon)$ in the 4d theory (see also section 6.1
below) and $g_{\alpha}\nu_{i}=O(\epsilon)$ in the 6d theory. Since
$\hat{F}_{k}\propto(g_{\alpha})^{k}$, it follows that, at the defect fixed
point, $\hat{F}_{k}\sim\epsilon^{k}$. More generally, it is of the form
$\hat{F}_{k}=\epsilon^{k}\,\sum_{r=0}a_{k,r}\,\epsilon^{r}$ . Thus, we see
that the last term in (4.16) contributes quadratically in $\epsilon$.
Consequently, the first two terms of $\hat{F}$ coincide with those of a
conformal theory.

Explicitly, to order $\epsilon^{0}$, $\epsilon^{1}$ and $\epsilon^{2}$, (4.16)
yields to

$\mu\frac{\partial\hat{F}_{0}}{\partial\mu}=0\,,\qquad\mu\frac{\partial\hat{F}_{1}}{\partial\mu}+\frac{\epsilon}{2}\hat{F}_{0}=0\,,\qquad\mu\frac{\partial\hat{F}_{2}}{\partial\mu}+\frac{\epsilon}{2}\hat{F}_{1}-c\,\epsilon\,\hat{F}_{1}=0\,.$
(4.17)

Appropriately redefining the integration constants the solution is

$\hat{F}=B+\frac{\epsilon}{2}\,\log\big{(}\frac{|\vec{p}_{T}|}{\mu}\big{)}+\frac{1-2c}{8}\,\epsilon^{2}\Big{(}\log\big{(}\frac{|\vec{p}_{T}|}{\mu}\big{)}\Big{)}^{2}+\frac{b_{1}}{2}(1-2c)\epsilon^{2}\log\big{(}\frac{|\vec{p}_{T}|}{\mu}\big{)}+\mathcal{O}(\epsilon^{3})\,,$
(4.18)

where $B=1+b_{1}\epsilon+b_{2}\epsilon^{2}$ and $b_{1}$, $b_{2}$ are
integration constants. It is convenient to factorize $B$, which then appears
in the overall normalization. Thus, to this order

$\hat{F}=1+\frac{\epsilon}{2}\,\log\big{(}\frac{|\vec{p}_{T}|}{\mu}\big{)}+\frac{1-2c}{8}\,\epsilon^{2}\Big{(}\log\big{(}\frac{|\vec{p}_{T}|}{\mu}\big{)}\Big{)}^{2}-b_{1}\,c\,\epsilon^{2}\log\big{(}\frac{|\vec{p}_{T}|}{\mu}\big{)}+\mathcal{O}(\epsilon^{3})\,.$
(4.19)

This reproduces our previous result (4.5), obtained by explicit calculation in
perturbation theory (with the integration constant $b_{1}=0$). In order to
compute higher order terms in the $\epsilon$ expansion, one would need to
compute $\beta_{i}$ to order $g_{\alpha}^{3}$.

As stressed earlier, the first two terms of $\hat{F}$ coincide with those of a
conformal theory. For this observable, the deviation from conformality is seen
at the order $O(\epsilon^{2})$.

##  5 RG flows in the Twins model

###  5.1 Fixed points and RG flows

As an application, here we study a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-symmetric model with two scalar fields in
$d=4-\epsilon$, defined by the action (hereafter, the “twins” model):

$\mathcal{S}=q\int
d^{d}x\left(\frac{1}{2}\big{(}\partial\phi_{1}\big{)}^{2}+\frac{1}{2}\big{(}\partial\phi_{2}\big{)}^{2}+V(\phi_{1},\phi_{2})\right)\,,$
(5.1)

with

$V(\phi_{1},\phi_{2})=\frac{1}{4}g_{1}\phi_{1}^{4}+\frac{1}{4}g_{2}\phi_{2}^{4}+\frac{1}{2}g_{3}\phi_{1}^{2}\phi_{2}^{2}\
.$ (5.2)

The model has a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry under
independent transformations $\phi_{1}\to-\phi_{1}$, and
$\phi_{2}\to-\phi_{2}$. When $g_{1}=g_{2}$ the theory is also symmetric under
the exchange $\phi_{1}\leftrightarrow\phi_{2}$. At the special point
$g_{1}=g_{2}=g_{3}$, the theory has $O(2)$ symmetry.

We assume that the potential is bounded from below, which requires $g_{1}\geq
0$, $g_{2}\geq 0$ and $g_{3}\geq-\sqrt{g_{1}g_{2}}$. In the present case, this
also implies that the potential is positive semi-definite. When
$g_{3}=\pm\sqrt{g_{1}g_{2}}$, the potential is a perfect square. If
$g_{3}=-\sqrt{g_{1}g_{2}}$, then the potential has flat directions
$\phi_{2}=\pm\phi_{1}(g_{1}/g_{2})^{\frac{1}{4}}$.

We now consider the action deformed by the defect,

$\mathcal{S}=q\int
d^{d}x\left(\frac{1}{2}\big{(}\partial\phi_{i}\big{)}^{2}+V(\phi_{i})+\nu_{i}\,\phi_{i}\,\delta_{T}(\vec{x})\right)\,,\quad
i=1,2\ .$ (5.3)

Just as in the general model (2.4) with $N$ scalar fields, in the $q\to\infty$
limit all bulk loop diagrams of the twins model are suppressed, leaving only
the diagrams computed in the previous sections. We can directly apply the
previous results to write down the $\beta$ functions and the VEV’s of
$\phi_{1}$ and $\phi_{2}$.

Setting $n=4$, $c=1$, from (3.19) we have

$\displaystyle\beta_{\nu_{1}}$ $\displaystyle=$
$\displaystyle\nu_{1R}\Big{(}-\frac{\epsilon}{2}+2\,\,\Omega\,(g_{1}(\nu_{1R})^{2}+g_{3}(\nu_{2R})^{2})$
(5.4) $\displaystyle-$ $\displaystyle
4\,\Omega^{2}\,\big{[}3g_{1}^{2}(\nu_{1R})^{4}+2g_{3}(2g_{1}+g_{3})(\nu_{1R})^{2}(\nu_{2R})^{2}+g_{3}(2g_{2}+g_{3})(\nu_{2R})^{4}\big{]}\Big{)}\,,$
$\displaystyle\beta_{\nu_{2}}$ $\displaystyle=$
$\displaystyle\nu_{2R}\Big{(}-\frac{\epsilon}{2}+2\,\,\Omega\,(g_{2}(\nu_{2R})^{2}+g_{3}(\nu_{1R})^{2})$
(5.5) $\displaystyle-$ $\displaystyle
4\,\Omega^{2}\,\big{[}3g_{2}^{2}(\nu_{2R})^{4}+2g_{3}(2g_{2}+g_{3})(\nu_{1R})^{2}(\nu_{2R})^{2}+g_{3}(2g_{1}+g_{3})(\nu_{1R})^{4}\big{]}\Big{)}\,.$

Let us first identify the fixed points. It is convenient to define

$x_{1}=\Omega\,g_{1}\,\nu_{1R}^{2}\ ,\quad x_{2}=\Omega\,g_{2}\,\nu_{2R}^{2}\
,\qquad\zeta=\frac{g_{3}}{g_{1}}\ ,\ \ \eta=\frac{g_{3}}{g_{2}}\ .$ (5.6)

Note that the parameters $\zeta$ and $\eta$ do not run, since
$g_{1},g_{2},g_{3}$ have the same classical RG flow, i.e.
$\beta_{g_{\alpha}}=-\epsilon g_{\alpha}$.

Fixed points can be divided into two classes. First of all, we have three
fixed points which do not depend on the parameters $\zeta,\,\eta$. They are
located at $(x_{1}^{*},x_{2}^{*})$ given by

$(x_{1}^{*},x_{2}^{*}):\ \qquad a)\ \,(0,\,0),\qquad b)\
\,(0,\,\frac{1}{4}\,(\epsilon+\frac{3}{2}\epsilon^{2}))\qquad b^{\prime})\
\,(\frac{1}{4}\,(\epsilon+\frac{3}{2}\epsilon^{2}),\,0)\,.$ (5.7)

In addition to these fixed points, there is a fixed point that depend on the
parameters, located at

$c)\
\,\left(\frac{1-\eta}{4(1-\zeta\eta)}\,(\epsilon+\frac{3}{2}\epsilon^{2}),\,\frac{1-\zeta}{4(1-\zeta\eta)}\,(\epsilon+\frac{3}{2}\epsilon^{2})\right)\
.$ (5.8)

It should be noted that existence of the fixed points requires $x_{1}^{*}\geq
0$ and $x_{2}^{*}\geq 0$ to ensure that the $\nu_{iR}$’s are real. Here we
assume $\epsilon>0$ (if $\epsilon<0$ there is only one fixed point at the
origin $(x_{1}^{*},x_{2}^{*})=(0,0)$ and is attractive). Note as well that in
the original $(\nu_{1R},\nu_{2R})$ variables, fixed points $b,\ b^{\prime}$
and $c$ give rise to a pair of “mirror” fixed points, defined by the two sign
choices $\nu_{iR}\propto\pm\sqrt{x_{i}}$.

Before further proceeding, let us pause to stress that in the double-scaling
limit we are taking the couplings in the original lagrangian are scaled as
$\mathfrak{g}_{\alpha}\to 0$ as $q^{-1}$. As a result, bulk loop diagrams are
suppressed (as expected for a semiclassical limit, since $q^{-1}$ plays the
role of $\hbar$). This is in particular reflected in the $\beta$ functions for
the bulk couplings, which are of the generic form
$\beta_{\mathfrak{g}_{\alpha}}\sim-\epsilon\,\mathfrak{g}_{\alpha}+O(\mathfrak{g}_{\alpha}^{3})$.
When rewritten in terms of the re-scaled couplings, these become
$\beta_{g_{\alpha}}\sim-\epsilon\,g_{\alpha}+O(\frac{g_{\alpha}^{3}}{q^{2}})\to\,-\epsilon\,g_{\alpha}$.
Therefore, the $\beta_{g_{\alpha}}$ functions are simply the classical ones:
in this limit, the bulk couplings run solely because of classical dimensional
analysis (in particular, note that in this limit one cannot sit in the WF
fixed point).999 An interesting problem is to incorporate further quantum
corrections in order to study the theory at both bulk and defect fixed points.
Demanding that $\beta_{g_{\alpha}}$ functions vanish, one has
$g_{\alpha}^{2}\sim\epsilon q^{2}$. Thus, at large $q$, bulk fixed points
would lie at strong coupling. A systematic organization of perturbation theory
may be attempted by choosing $\epsilon\sim 1/q^{2}$. We leave this problem for
future studies. Thus, while the fixed points occur at small values
$g\nu^{2}=\mathfrak{g}\mathfrak{h}^{2}=O(\epsilon)$, bulk loop corrections
come with extra powers of $q^{-1}$, and thus are negligible with respect to
the diagrams that we have considered.101010 The double-scaling limit selects
the same diagrams as the “ladder” approximation in [15, 17].

IR stability can be studied by linearization about a given fixed point. The
defect RG flow is a gradient (in the $\nu_{iR}$’s) flow of the “energy
function” $-\mathcal{H}$ when one uses the RG time variable $t=-\log\mu$.
Computing the eigenvalues $(\lambda_{1},\lambda_{2})$ of the Hessian at each
fixed point one finds (here we work to order $\epsilon$)

$(\lambda_{1},\lambda_{2}):\qquad a)\,\
(\frac{\epsilon}{2},\,\frac{\epsilon}{2}),\qquad b)\,\
(-\epsilon,\,\frac{\epsilon}{2}(1-\eta)),\qquad b^{\prime})\,\
(-\epsilon,\,\frac{\epsilon}{2}(1-\zeta))\,,$ (5.9)

and

$c)\,\ (-\epsilon\,\frac{(1-\eta)\,(1-\zeta)}{1-\eta\,\zeta},\,-\epsilon)\,.$
(5.10)

A given fixed point is IR stable in all directions if both eigenvalues are
negative and it develops an unstable direction when an eigenvalue becomes
positive. Note in particular that the origin, i.e. fixed point $a$, is always
repulsive since we are assuming $\epsilon>0$.

There are three different regimes:

  1. (i)

$\zeta,\,\eta<1$. Then $c$ is attractive, whereas $b$ and $b^{\prime}$ are
repulsive.

  2. (ii)

$\zeta<1$, $\eta>1$. Here $b$ is an attractor and $b^{\prime}$ is repulsive.
There is no fixed point $c$ in this regime, since this exists provided
$(1-\eta)\,(1-\zeta)>0$. [The case $\zeta>1$, $\eta<1$ is equivalent under the
exchange of $b$ and $b^{\prime}$.]

  3. (iii)

$\zeta,\,\eta>1$. $b$ and $b^{\prime}$ are attractors and $c$ is repulsive.

In fig. 3 we show the RG flows in each of these regimes. Note that regime (i)
also includes the cases $\zeta<0$ and $\eta<0$, corresponding to $g_{3}<0$.

i A case with $\eta<1$, $\zeta<1$. Here
$(\zeta,\eta)=(\frac{1}{2},\frac{1}{3})$ .

ii $(\zeta,\eta)=(2,\frac{1}{3})$, describing a case with $\eta<1$, $\zeta>1$.

iii $(\zeta,\eta)=(2,3)$, describing a case with $\eta>1$, $\zeta>1$.

Figure 3: RG trajectories for representatives of each relevant case. Blue,
green and red dots represent, respectively, the $a$, $\\{b,\,b^{\prime}\\}$
and $c$ fixed points.

Interestingly, as one goes from one regime to another, the fixed point $c$ may
disappear or reappear. This happens through fixed point annihilation/creation
between two $c$ “mirror” fixed points. The annihilation/creation occurs either
at $\nu_{1R}=0$ or $\nu_{2R}=0$, that is, when either $\eta=1$ or $\zeta=1$.
At these critical values of the parameters, the fixed point $c$ merges either
with $b$ or with $b^{\prime}$. In fig. 4 we show an example of this phenomenon
when going from (i) to (ii).

Figure 4: Fixed point annihilation as $\zeta$ grows for fixed $\eta=1/3$. When
$\zeta>1$, the RG flow is as illustrated in fig. 3 (ii).

The twins model interpolates between the $O(2)$ model and the two decoupled
$\phi^{4}$ theories that arise when $g_{3}\to 0$. The latter has
$\eta=\zeta=0$ and the familiar WF fixed points, which agree with (5.7) and
(5.8), taking into account that there are two possible solutions for each
$x_{i}^{*}$, the trivial fixed point at the origin and the WF fixed point.

On the other hand, in the $O(2)$ limit, $\eta\to 1$, $\zeta\to 1$, the fixed
point $c)$ is singular and its limiting value is ambiguous as it depends on
the path the point $\eta=\zeta=1$ is approached. The general solution to
$\beta_{1}=\beta_{2}=0$ is given by a circle of fixed points at $\Omega
g_{1}(\nu_{1R}^{2}+\nu_{2R}^{2})=\frac{1}{4}\epsilon+\frac{3}{8}\epsilon^{2}+\cdots$.

###  5.2 Fixed points to all orders: A Conjecture

A striking property of the fixed point $c)$, computed to $O(g^{2})$ in (5.8),
is that the $\epsilon$ dependence factorizes from the coupling dependence. We
will call this phenomenon “dimensional disentangling”. A natural question is
whether this property extends to higher orders or it is just an accident of
the first two orders. Factorization to all orders may seem like a miracle,
since at high orders there are a large number of Feynman diagrams involving
$g_{1},g_{2}$ and $g_{3}$ in a non-trivial way. Nevertheless, as we explain
below, there are strong indications that this factorization property must be
an exact feature of the large $q$ limit, which is maintained to all order in
$\epsilon$. Concretely, we conjecture that the fixed point $c$ is exactly
located at

$c):\quad(x_{1}^{*},x_{2}^{*})=\left(\frac{1-\eta}{4(1-\zeta\eta)}\,f(\epsilon),\,\frac{1-\zeta}{4(1-\zeta\eta)}\,f(\epsilon)\right)\
,$ (5.11)

and, to all orders, there are other three fixed points $a$, $b$ and
$b^{\prime}$ located at

$a)\ \,(0,\,0),\qquad b)\ \,\big{(}0,\,\frac{1}{4}\,f(\epsilon)\big{)}\
,\qquad b^{\prime})\ \,\big{(}\frac{1}{4}\,f(\epsilon),\,0\big{)}\,.$ (5.12)

To justify this conjecture, let us first consider the fixed points $b$ and
$b^{\prime}$. When either $\nu_{1R}=0$ or $\nu_{2R}=0$, there is only one non-
trivial $\beta$ function. This must be identical to the quartic model with a
single coupling. Consider, for example, the case $\nu_{2R}=0$. Because bulk
loops are suppressed, there is no Feynman diagram involving vertices that
depend on $g_{2}$ or $g_{3}$, since any $\phi_{2}$ line could only appear in
loops. Therefore the $\beta$ function must be of the form
$\beta_{1}=\nu_{1R}\,[-\frac{\epsilon}{2}+P(x_{1})]$, which ensures that
$x_{1}^{*}=f(\epsilon)$.

On the other hand, the fixed point $c$ must merge with fixed points $b$ or
$b^{\prime}$ when fixed point annihilation between mirror fixed points occur,
since (due to the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry) this always
occurs either at $\nu_{1R}=0$ or $\nu_{2R}=0$, and $b$ or $b^{\prime}$ are the
unique solutions with either $\nu_{1R}=0$ or $\nu_{2R}=0$ (in perturbation
theory in $\epsilon$). This implies that, if $c$ is of the form (5.11),
exactly the same function $f(\epsilon)$ must appear in the fixed points $b,\
b^{\prime}$.

In what follows we present more evidence for this conjecture: we will now
explicitly show that factorization holds at order $g^{3}$, $g^{4}$ and
$g^{5}$.

Consider computing computing the 1-point function in terms of Feynman
diagrams. The Feynman rules for the vertices are schematically as shown in
fig. 5 below.

Figure 5: Feynman rules for the vertices (each function is evaluated on the
renormalized couplings). Black circles represent the attachment to the defect
while squares are bulk vertices. In addition, by virtue of the double-scaling
limit, bulk loops are not allowed and surviving diagrams have the largest
number of defect insertions (that is, black circles).

With these rules, one may can construct all possible diagrams to a given order
in the couplings (equivalently, in $V^{R}$). An extra piece of information is
that the RG flow must be a gradient flow of the $\mathcal{H}$ function
constructed from $V$ and its derivatives evaluated on the renormalized
couplings. This makes it possible to further restrict possible contributions
to $\mathcal{H}$, which, to fifth order, can only contain the following terms:

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{4}\,\nu_{iR}^{2}+2\,\Omega\,V^{R}-2\,\Omega^{2}\,(V_{i}^{R})^{2}$
$\displaystyle+\alpha_{1}\,\Omega^{3}\,V_{ij}^{R}\,V_{i}^{R}\,V_{j}^{R}$
$\displaystyle+\alpha_{2}\,\Omega^{4}\,V_{ijk}^{R}\,V_{i}^{R}\,V_{j}^{R}\,V_{k}^{R}+\alpha_{3}\,\Omega^{4}\,V_{ij}^{R}\,V_{jk}^{R}\,V_{k}^{R}\,V_{i}^{R}$
$\displaystyle+\alpha_{4}\,\Omega^{5}\,V_{ijkl}^{R}\,V_{i}^{R}\,V_{j}^{R}\,V_{k}^{R}\,V_{l}^{R}+\alpha_{5}\,\Omega^{5}\,V_{ijk}^{R}\,V_{kl}^{R}\,V_{i}^{R}\,V_{j}^{R}\,V_{l}^{R}+\alpha_{6}\,\Omega^{5}\,V_{ij}^{R}\,V_{jk}^{R}\,V_{kl}^{R}\,V_{l}^{R}\,V_{i}^{R}$
$\displaystyle+\cdots\,,$

where $\alpha_{i}$ are numerical coefficients which can be determined from
higher loop integrals. Using that $\beta_{i}=2\partial_{i}\mathcal{H}$, and
solving $\beta_{1}=\beta_{2}=0$ in perturbation theory, we find that fixed
points $a$, $b$, $b^{\prime}$ still exist and are located at

$a)\ \,(0,\,0),\qquad b)\ \,\big{(}0,\,\frac{1}{4}\,f(\epsilon)\big{)}\
,\qquad b^{\prime})\ \,\big{(}\frac{1}{4}\,f(\epsilon),\,0\big{)}\,,$ (5.14)

while the $c$ fixed point sits at

$c)\
\,\left(\frac{1-\eta}{4(1-\zeta\eta)}\,f(\epsilon),\,\frac{1-\zeta}{4(1-\zeta\eta)}\,f(\epsilon)\right)\
,$ (5.15)

with

$\displaystyle f(\epsilon)$ $\displaystyle=$
$\displaystyle\epsilon+\frac{3}{2}\epsilon^{2}+\frac{3}{4}(6-\alpha_{1})\,\epsilon^{3}+\frac{15}{64}(72-24\alpha_{1}-2\alpha_{2}-3\alpha_{3})\,\epsilon^{4}$
$\displaystyle+\frac{9}{128}(1008-504\alpha_{1}+24\alpha_{1}^{2}-60\alpha_{2}-90\alpha_{3}-2\alpha_{4}-6\alpha_{5}-9\alpha_{6})\,\epsilon^{5}\cdots\,,$

Therefore, coupling dependence and $\epsilon$ dependence still factorize to
orders $g^{3}$, $g^{4}$ and $g^{5}$. This strongly indicates that this
remarkable property must hold to all orders.

Finally, it may be worth noting that the factorization property allows one to
define $\epsilon^{\prime}\equiv f(\epsilon)$ so that
$d=4-\epsilon^{\prime}+O\big{(}{\epsilon^{\prime}}^{2}\big{)}$ and the fixed
points become of the form

$b):\ (0,\frac{1}{4}\epsilon^{\prime})\ ,\quad b):\
(0,\frac{1}{4}\epsilon^{\prime})\ ,\quad c)\
\,\left(\frac{1-\eta}{4(1-\zeta\eta)}\,\epsilon^{\prime},\,\frac{1-\zeta}{4(1-\zeta\eta)}\,\epsilon^{\prime}\right)\
.$

All higher order corrections get encapsulated in $4-d$ (given by
$f^{-1}(\epsilon^{\prime})$). This redefinition of $\epsilon$ is
straightforward if $f(\epsilon)$ is monotonic. A non-monotonic $f(\epsilon)$
could give rise to interesting effects at finite values of $\epsilon$.

##  6 Discussion

Let us summarize the main results of this paper. We considered a theory with
$N$ interacting scalar fields with a general (marginal) potential in
$d=4-\epsilon$ and $d=6-\epsilon$, deformed by localized defect (a line in 4d;
a surface in 6d). The deformation is implemented by an operator which is a
linear combination of the scalar fields. We showed that there is a double-
scaling limit where the bulk theory is classical –bulk loops are suppressed–
whereas the defect theory is quantum. This generalizes earlier related studies
[4, 5, 8].

The renormalization of the defect theory was here carried out explicitly up to
order $g_{\alpha}^{2}$, where the $g_{\alpha}$’s represent the bulk couplings.
We have computed the $\beta$ functions of the defect couplings and the VEV’s
of the scalar fields on the defect. We have shown that the $\epsilon$
dependence and general structure of the VEV’s at the fixed points are dictated
by the renormalization group, by providing an independent derivation through
the Callan-Symanzik equation.

There are a number of extremely interesting open problems:

  1. 1.

In section 3, we have found that, to two-loops in the defect couplings, the
$\beta$-function of the defect couplings can be written as a gradient,
$\partial_{i}\beta=2c\partial_{i}\mathcal{H}$, for both the case of a line
defect in $d=4-\epsilon$ dimensions and the case of a surface defect in
$d=6-\epsilon$ dimensions. It would be very interesting to prove more
generally that the defect $\beta$ functions must be the gradient of a scalar
function.

  2. 2.

For the case of line defects in $d=4-\epsilon$ dimensions, it has been shown
in section 3.1 that the function $\exp(\mathcal{H})$ is the VEV of the
circular defect, in agreement with [2, 5, 15]. Similarly, it would be
extremely interesting to elucidate the interpretation of the $\mathcal{H}$
function in $d=6-\epsilon$ dimensions as, perhaps, minus the on-shell action
of a spherical defect (or other closed surface).

  3. 3.

As shown in section 4, the non-conformality of the bulk theory kicks in at
two-loops in the defect couplings in the one-point function. On the face of
it, this suggests that the identification of $\exp(\mathcal{H})$ with the VEV
of the circular defect may fail starting from two loops. It would be
interesting to investigate this further.

  4. 4.

The RG flows of the defect couplings depend on RG invariant parameters given
by the ratio of bulk couplings. This generically leads to an interesting
pattern of fixed points which move, annihilate or are created, as the bulk
couplings are varied. This was illustrated by considering a line defect in a
simple $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetric model with two scalar
fields –the “twins model”. The model interpolates between the $O(2)$ theory
and two decoupled $\phi^{4}$ theories, and it can be viewed as the $O(2)$
theory with the addition of a general $O(2)$ breaking deformation preserving
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. A problem of interest is to investigate
the effects of fixed point creation/annihilation in other observables, such as
e.g. correlation functions (for which recent results have been obtained [5,
22]) of defect operators or fusion of defects (such as in [23, 8]).

  5. 5.

The fixed points exhibit a remarkable structure where the
$\epsilon$-dependence factorizes from the coupling dependence. We argued that
dimensional disentangling is a consequence of the double scaling limit and
that it must hold to all orders. A corollary of this hypothesis is that the
location of fixed points and the properties of the RG flows determined at one
loop are the same to all orders in the $\epsilon$ expansion, modulo an overall
change of scale $\epsilon\to f(\epsilon)$. An open question is to understand
the underlying reason for this factorization and whether this property still
holds for theories with $N$ scalar fields (a preliminary calculation for three
scalar fields suggests that the factorized form of fixed points still holds in
this case). It would also be very interesting to see if dimensional
disentangling also holds in more general defect theories, in particular, gauge
theories with defects, upon taking a suitable large defect coupling limit.

## Acknowledgements

D.R-G. thanks the CERN Theory Division for warm hospitality as this work was
being finished. He also thanks the organizers of the Large charge at Les
Diablerets meeting (as well as to SwissMAP) for a very stimulating environment
where part of this work was carried out. D.R-G is partially supported by the
Spanish government grant MINECO-16-FPA2015-63667-P. He also acknowledges
support from the Principado de Asturias through the grant FC-GRUPIN-
IDI/2018/000174. J.G.R. acknowledges financial support from projects MINECO
grant PID2019-105614GB-C21, and from the State Agency for Research of the
Spanish Ministry of Science and Innovation through the “Unit of Excellence
María de Maeztu 2020-2023” (CEX2019-000918-M).

##  Appendix A Some properties of the one-point function

In this appendix we will review some basic aspects about 1-point functions in
the presence of a defect. As we have argued in the main text, the violation of
conformal invariance in the bulk due to the non-zero $\beta$ function of the
bulk coupling only enters at the two-loop order. Thus, to one-loop order we
effectively have a defect CFT (dCFT).

On general grounds, in a dCFT, we expect a bulk-defect OPE, which allows us to
write bulk operators $O$ in terms of defect operators $\mathcal{O}$ (see e.g.
[1]),

$O=\sum_{\mathcal{O}\in{\rm
defect}}\,c_{O\mathcal{O}}\,\frac{\mathcal{O}}{|\vec{x}_{T}|^{\Delta_{O}-\Delta_{\mathcal{O}}}}\,.$
(A.1)

We recall that $d_{T}=d-d_{\rm wv}$, so $d_{T}=3-\epsilon$ in the 4d theory
and $d_{T}=4-\epsilon$ in the 6d theory. Since the defect is at a fixed point,
only the identity operator can take a VEV. As a consequence, it immediately
follows upon taking a VEV that the bulk 1-point function is

$\langle O(\vec{x}_{T})\rangle=\frac{c_{O}}{|\vec{x}_{T}|^{\Delta_{O}}}\,.$
(A.2)

This formula can be explicitly verified by direct computation. Consider the
computation of the 1-point function of the bulk field from the path integral.
In the saddle-point approximation, this boils down to evaluating the integrand
in the saddle-point solution as

$\langle\phi_{i}\rangle=\int\mathcal{D}\phi\,\phi_{i}\,e^{-S}=\phi_{i}\Big{|}_{\rm
saddle}=\phi_{i}^{(0)}+\phi_{i}^{(1)}\,,$ (A.3)

where $\phi_{i}^{(0)},\,\phi_{i}^{(1)}$ stand for the saddle-point solutions
in (2.9). As an example, let us focus on the leading term,

$\langle\phi_{i}\rangle=-\nu_{i}\,\int d^{d}z\,G(x-z)\,\delta_{T}(z)\,.$ (A.4)

Inserting the propagator explicitly, we get

$\langle\phi_{i}\rangle=-\nu_{i}\,\int
d^{d}z\,\int\frac{d^{d}p}{(2\pi)^{d}}\,\frac{e^{ip(x-z)}}{p^{2}}\,\delta_{T}(z)\,.$
(A.5)

Therefore

$\langle\phi_{i}\rangle=-\nu_{i}\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{e^{i\vec{p}_{T}\cdot\vec{x}_{T}}}{|\vec{p}_{T}|^{2}}\,.$
(A.6)

Next, performing the Fourier transform, we obtain

$\langle\phi\rangle\sim-\frac{\nu}{|x_{T}|^{d_{T}-2}}=\begin{cases}-\frac{\nu}{|x_{T}|}\,,\quad\
d=4,\,d_{T}=3\ ,\\\ -\frac{\nu}{|x_{T}|^{2}}\,,\quad d=6,\,d_{T}=4\
;\end{cases}$ (A.7)

which precisely reproduces (A.2).

It is also easy to make contact with the results in the main text. Indeed,
consider integrating the 1-point function above against a delta function in
the transverse space. A simple computation shows that

$\langle\int
d^{d}x\,\phi\,\delta_{T}(\vec{x})\rangle=-\nu\,\mathcal{V}\,\int\frac{d\vec{p}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{|\vec{p}_{T}|^{2}}\,.$
(A.8)

where $\mathcal{V}=\int dx_{||}$ is the regularized defect volume. This
reproduces the leading term in (3.3).

The next contributions in the $\epsilon$ expansion can be computed
systematically, similarly as we did in sections 2–4 (see [8]). In particular,
in $d=4$, for a dCFT (that is, a theory sitting in bulk and defect fixed
points) one still obtains (A.2) with $\Delta_{\phi}$ given by an expansion of
the form (see e.g. [4])
$\Delta_{\phi}=1-\frac{1}{2}\epsilon+b\epsilon^{2}+...$. In momentum space,
this is

$\langle\tilde{\phi}(p)\rangle\sim-\frac{{\cal
N}}{|p_{T}|^{2-\epsilon/2-b\epsilon^{2}+...}}\ .$ (A.9)

To $O(\epsilon)$, this is indeed the form of the one-point function (4.5).
However, in the present case, as described in section 4, at the two-loop order
the effect of the non-zero bulk $\beta$ function kicks in. As a consequence,
the one-point function (4.5) to $O(\epsilon^{2})$ does not have the form
(A.9). While there is still a bulk-defect OPE, the $c_{O\mathcal{O}}$ now
become functions of the dimensionless combination $\mu\,|\vec{x}_{T}|$. Thus,
instead of (A.2), one now finds that

$\langle
O(\vec{x}_{T})\rangle=\frac{c_{O}}{|\vec{x}_{T}|^{\Delta_{O}}}\,f(\mu\,|\vec{x}_{T}|)\,.$
(A.10)

As shown in section 4.1, the form of $f$ is constrained by the fact that the
1-point function is subject to a Callan-Symanzik equation. We have shown that
this constrained form of $f$ agrees with the $f$ obtained independently by
direct calculation in terms of Feynman diagrams.

##  Appendix B Integrals in $d=4-\epsilon$

In this subsection, we compute the $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$
integrals in $d=4-\epsilon$, where $n=4$.

###  B.1 The $\mathcal{I}_{1}$ integral

When $n=4$, the integral $\mathcal{I}_{1}$ is given by

$\mathcal{I}_{1}=\int\frac{d\vec{k}_{T}^{1}}{(2\pi)^{d_{T}}}\int\frac{d\vec{k}_{T}^{2}}{(2\pi)^{d_{T}}}\frac{1}{(\vec{k}_{T}^{1})^{2}\,(\vec{k}^{2}_{T})^{2}\,(\vec{p}_{T}-\vec{k}_{T}^{1}-\vec{k}_{T}^{2})^{2}}\,.$
(B.1)

This integral has been recently computed in eq.(114) in [8], using the general
formulas of [24]. One finds

$\int\frac{d\vec{k}_{T}^{1}}{(2\pi)^{d_{T}}}\,\int\frac{d\vec{k}_{T}^{2}}{(2\pi)^{d_{T}}}\,\frac{1}{(\vec{k}_{T}^{1})^{2}\,(\vec{k}_{T}^{2})^{2}\,(\vec{k}_{T}^{3}-\vec{k}_{T}^{1}-\vec{k}_{T}^{2})^{2}}=\frac{\pi^{d-1}}{(2\pi)^{2d-2}}\,G(1,1)\,G(1,\frac{5-d}{2})\,|\vec{p}_{T}|^{2d-8}\,.$
(B.2)

where

$G(n_{1},n_{2})\equiv\frac{\Gamma(n_{1}+n_{2}-d_{T}/2)\Gamma(d_{T}/2-n_{1})\Gamma(d_{T}/2-n_{2})}{\Gamma(n_{1})\Gamma(n_{2})\Gamma(d_{T}-n_{1}-n_{2})}\
,\qquad d_{T}=d-d_{\rm wv}\ ,$ (B.3)

with $d_{\rm wv}=1$ in the 4d theory and $d_{\rm wv}=2$ in the 6d theory.
Expanding at small $\epsilon$, we obtain

$\mathcal{I}_{1}=\frac{\Omega}{\epsilon}-2\,\Omega\,\log|\vec{p}_{T}|+A\epsilon+2\Omega\epsilon\,(\log|\vec{p}_{T}|)^{2}\,,\qquad
A\equiv\frac{9\,\Omega}{2}-\frac{7}{768}\ .$ (B.4)

We have omitted the dependence on $\mu$, which can be restored by
$|\vec{p}_{T}|\to|\vec{p}_{T}|/\mu$. We have also absorbed a numerical
constant in the definition of $\mu$ by the rescaling
$\mu^{-1}\,|\vec{p}_{T}|\rightarrow\mu^{-1}\,|\vec{p}_{T}|\,e^{\frac{1}{2}(3-\gamma_{E}+\log(4\pi))}$.

###  B.2 The $\mathcal{I}_{2}$ integral

For $\mathcal{I}_{2}$ we have

$\displaystyle\mathcal{I}_{2}$ $\displaystyle=$
$\displaystyle\int\frac{d\vec{k}_{T}^{1}}{(2\pi)^{d_{T}}}\,\int\frac{d\vec{k}_{T}^{2}}{(2\pi)^{d_{T}}}\,\int\frac{d\vec{k}_{T}^{3}}{(2\pi)^{d_{T}}}\,\int\frac{d\vec{k}_{T}^{4}}{(2\pi)^{d_{T}}}$
(B.5)
$\displaystyle\frac{1}{(\vec{k}_{T}^{1})^{2}\,(\vec{k}_{T}^{2})^{2}\,(\vec{k}_{T}^{3})^{2}\,(\vec{k}_{T}^{4})^{2}\,(\vec{k}_{T}^{3}-\vec{k}_{T}^{1}-\vec{k}_{T}^{2})^{2}\,(\vec{p}_{T}-\vec{k}_{T}^{3}-\vec{k}_{T}^{4})^{2}}\,.$

This can be written as

$\displaystyle\mathcal{I}_{2}=\int\frac{d\vec{k}_{T}^{3}}{(2\pi)^{d_{T}}}\,\frac{1}{(\vec{k}_{T}^{3})^{2}}\,$
$\displaystyle\Big{[}\int\frac{d\vec{k}_{T}^{1}}{(2\pi)^{d_{T}}}\,\int\frac{d\vec{k}_{T}^{2}}{(2\pi)^{d_{T}}}\,\frac{1}{(\vec{k}_{T}^{1})^{2}\,(\vec{k}_{T}^{2})^{2}\,(\vec{k}_{T}^{3}-\vec{k}_{T}^{1}-\vec{k}_{T}^{2})^{2}}\Big{]}\,\Big{[}\int\frac{d\vec{k}_{T}^{4}}{(2\pi)^{d_{T}}}\frac{1}{(\vec{k}_{T}^{4})^{2}\,(\vec{p}_{T}-\vec{k}_{T}^{3}-\vec{k}_{T}^{4})^{2}}\Big{]}\,.$

From eq.(114) in [8] (see also [24])

$\int\frac{d\vec{k}_{T}^{1}}{(2\pi)^{d_{T}}}\,\int\frac{d\vec{k}_{T}^{2}}{(2\pi)^{d_{T}}}\,\frac{1}{(\vec{k}_{T}^{1})^{2}\,(\vec{k}_{T}^{2})^{2}\,(\vec{k}_{T}^{3}-\vec{k}_{T}^{1}-\vec{k}_{T}^{2})^{2}}=\frac{\pi^{d-1}}{(2\pi)^{2d-2}}\,|\vec{k}_{T}^{3}|^{2(d-4)}\,G(1,1)\,G(1,\frac{5-d}{2})\,.$
(B.6)

In turn the second integral can be read off from eq.(123) in [8],

$\int\frac{d\vec{k}_{T}^{4}}{(2\pi)^{d_{T}}}\frac{1}{(\vec{k}_{T}^{4})^{2}\,(\vec{p}_{T}-\vec{k}_{T}^{3}-\vec{k}_{T}^{4})^{2}}=\frac{2^{5-2d}\,\pi^{2-\frac{d}{2}}}{\cos\big{(}\frac{d\pi}{2}\big{)}\,\Gamma\big{(}\frac{d-2}{2}\big{)}}\frac{1}{(\vec{p}_{T}-\vec{k}_{T}^{3})^{5-d}}\,.$
(B.7)

Thus we need to compute

$\mathcal{I}_{2}=\frac{2^{7-4d}\,\pi^{3-\frac{3d}{2}}}{\cos\big{(}\frac{d\pi}{2}\big{)}\,\Gamma\big{(}\frac{d-2}{2}\big{)}}\,G(1,1)\,G(1,\frac{5-d}{2})\,\int\frac{d\vec{k}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{(\vec{k}_{T})^{2\,(5-d)}\,(\vec{k}_{T}-\vec{p})^{5-d}}\,$
(B.8)

Using [24]

$\int\frac{d\vec{k}_{T}}{(2\pi)^{d_{T}}}\,\frac{1}{(\vec{k}_{T})^{2\,(5-d)}\,(\vec{k}_{T}-\vec{p})^{5-d}}=\frac{\pi^{\frac{d-1}{2}}}{(2\pi)^{d-1}}\,|\vec{p}_{T}|^{4\,(d-4)}\,G(5-d,\frac{5-d}{2})\,$
(B.9)

Combining the above expressions, we get

$\mathcal{I}_{2}=\frac{2^{8-5d}\,\pi^{\frac{7}{2}-2d}}{\cos\big{(}\frac{d\pi}{2}\big{)}\,\Gamma\big{(}\frac{d-2}{2}\big{)}}\,G(1,1)\,G(1,\frac{5-d}{2})\,G(5-d,\frac{5-d}{2})\,|\vec{p}_{T}|^{4(d-4)}\,.$
(B.10)

Expanding in powers of $\epsilon$, we find

$\mathcal{I}_{2}=\frac{\Omega^{2}}{2\,\epsilon^{2}}+\frac{\Omega^{2}\,(1-2\log|\vec{p}_{T}|)}{\epsilon}+\Big{(}C-4\,\Omega^{2}\,\log|\vec{p}_{T}|+4\,\Omega^{2}\,(\log|\vec{p}_{T}|)^{2}\Big{)}\,.$
(B.11)

where $C=\frac{19\,\Omega^{2}}{2}-\frac{11\,\Omega}{768}$ and we have rescaled
$\mu$ by the same numerical constant as in (B.4).

##  Appendix C Integrals in $d=6-\epsilon$

We now compute the $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ integrals near
$d=6$, where $n=3$.

###  C.1 The $\mathcal{I}_{1}$ integral

For $\mathcal{I}_{1}$ we have

$\mathcal{I}_{1}=\int\frac{d\vec{k}_{T}}{(2\pi)^{d_{T}}}\frac{1}{\vec{k}_{T}^{2}\,(\vec{p}_{T}-\vec{k}_{T})^{2}}\,.$
(C.1)

We can directly borrow (123) in [8]. In $6-\epsilon$ dimensions,

$\mathcal{I}_{1}=\frac{\Omega}{\epsilon}-\Omega\,\log|\vec{p}_{T}|+A\epsilon+\epsilon\,\frac{\Omega}{2}\,(\log|\vec{p}_{T}|)^{2}\,,\qquad
A\equiv\frac{\Omega}{2}-\frac{1}{384}\ .$ (C.2)

Again, we have omitted the dependence on $\mu$, which is restored by
$|\vec{p}_{T}|\to|\vec{p}_{T}|/\mu$, and absorbed a numerical constant in the
definition of $\mu$ by
$\mu^{-1}\,|\vec{p}_{T}|\rightarrow\mu^{-1}\,|\vec{p}_{T}|\,e^{\frac{1}{2}(2-\gamma_{E}+\log(4\pi))}$.

###  C.2 The $\mathcal{I}_{2}$ integral

In the 6d model, the $\mathcal{I}_{2}$ integral is given by

$\mathcal{I}_{2}=\int\frac{d\vec{k}_{T}^{1}}{(2\pi)^{d_{T}}}\int\frac{d\vec{k}_{T}^{2}}{(2\pi)^{d_{T}}}\frac{1}{(\vec{k}_{T}^{1})^{2}\,(\vec{k}_{T}^{2})^{2}\,(\vec{k}_{T}^{2}-\vec{k}_{T}^{1})^{2}\,(\vec{k}_{T}^{2}-\vec{p}_{T})^{2}}\,.$
(C.3)

This integral is a particular case of the integral computed in eq. (2.8) in
[24], with $n_{1}=0$, $n_{2}=1$, $n_{3}=1$, $n_{4}=1$, $n_{5}=1$. This gives

$\mathcal{I}_{2}=\frac{\pi^{d-2}}{(2\pi)^{2(d-2)}}\,|\vec{p}_{T}|^{2d-12}\,G(1,1)\,G(1,\frac{8-d}{2})\,.$
(C.4)

Expanding in powers of $\epsilon$, we obtain

$\mathcal{I}_{2}=\frac{\Omega^{2}}{2\epsilon^{2}}+\frac{\Omega^{2}\,(1-4\,\log|\vec{p}_{T}|)}{4\epsilon}+\Big{(}C-\frac{1}{2}\Omega^{2}\,\log|\vec{p}_{T}|+\Omega^{2}\,(\log|\vec{p}_{T}|)^{2}\Big{)}\,.$
(C.5)

where, in the 6d model, we define
$C\equiv\frac{7\Omega^{2}}{8}-\frac{\Omega}{384}$. We have rescaled $\mu$ by
the same numerical constant as in (C.2).

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