###### Abstract

Conformal symmetry can be spontaneously broken due to the presence of a defect
or other background, which gives a symmetry-breaking vacuum expectation value
(VEV) to some scalar operators. We study the effective field theory of
fluctuations around these backgrounds, showing that it organizes as an
expansion in powers of the inverse of the VEV, and computing some of the
leading corrections. We focus on the case of space-like defects in a four-
dimensional Lorentzian theory relevant to the pseudo-conformal universe
scenario, although the conclusions extend to other kinds of defects and to the
breaking of conformal symmetry to Poincaré symmetry.

On the EFT of Conformal Symmetry Breaking

Kurt<EMAIL_ADDRESS>Qiuyue
<EMAIL_ADDRESS>Mark<EMAIL_ADDRESS>
aCERCA, Department of Physics,

Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106, USA

bCenter for Particle Cosmology, Department of Physics and Astronomy,

University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

###### Contents

  1. 1 Introduction
  2. 2 The EFT of conformal symmetry
    1. 2.1 Direct construction
    2. 2.2 Comparison to the coset construction
  3. 3 Breaking to Poincaré
    1. 3.1 2-point function
  4. 4 Defect CFT breaking
  5. 5 Defect CFT 2-point function
    1. 5.1 Leading part
    2. 5.2 Next to leading order corrections
    3. 5.3 Two-point reducible diagram
    4. 5.4 Cubic loop
  6. 6 Conclusions
  7. A Details of the contour choice

##  1 Introduction

In the real world, no system is infinite, and boundary effects inevitably
become important. Therefore, boundaries and defects in quantum field theory
are natural subjects of interest in studying systems with finite extent or
where multiple regions or phases are joined by junctions or localized
impurities. In particular, recent years have seen increased activity in the
study of such boundaries and defects in the context of conformal field theory
(CFT) (see e.g. [1, 2, 3] for overviews.)

A defect in a CFT can be of any dimension. One of the simplest cases is when
the defect is straight. A straight $d$ dimensional defect breaks the $D$
dimensional conformal group down to the $d$ dimensional conformal group, plus
the group of rotations around the defect. In a Lorentzian CFT, such a defect
can be space-like or time-like, but the more frequently-discussed case is that
of a time-like defect, where the defect represents a spatial boundary or
interface in the system.

In contrast, an example where space-like defects are relevant is in some non-
inflationary early universe scenarios featuring a pre-big bang phase, in which
the universe is nearly flat rather than accelerating. Some well-known examples
of this type are the ekpyrotic scenario [4, 5] and genesis-type models [6, 7].
The general class of such models that makes use of a space-like conformal
defect is the pseudo-conformal universe scenario [8, 7, 9, 10, 11, 12, 13, 14,
15]. In these scenarios, it is postulated that the universe before reheating
is described by a CFT on a nearly Minkowski spacetime whose conformal algebra
is broken spontaneously by a time-dependent vacuum expectation value (VEV) of
some dimension $\Delta$ scalar primary operator $\Phi$ taking the form

$\langle\Phi\rangle={C^{\Delta}\over{(-t)}^{\Delta}}\ ,$ (1.1)

where the dimensionless constant $C$ signals the strength of the symmetry
breaking. The VEV (1.1) breaks the four-dimensional Lorentzian conformal
symmetry down to a three-dimensional Euclidean conformal symmetry

$\mathfrak{so}(4,2)\rightarrow\mathfrak{so}(4,1)\ .$ (1.2)

As $t\rightarrow 0$ from below, the VEV (1.1) diverges and the universe then
reheats and transitions to the standard post-big bang radiation domination
phase. The reheating surface at $t=0$ is the space-like defect in the CFT.
This CFT scenario can also be a given a five-dimensional AdS dual description
[16, 17, 18, 19, 20]

Our primary interest here will be in studying the effective field theory (EFT)
that describes fluctuations around the symmetry breaking vacuum described by
(1.1). In [11] an effective field theory for studying such fluctuations was
described. Here we will further explore some of the systematics of this
effective theory. In particular, we will see how the EFT expansion organizes
itself as a power series in various powers of $1/C$, and we will compute some
of the leading corrections to the 2-point function.

Since our original motivation came from studying the pseudo-conformal universe
scenario, we will specialize in this paper to the case of a space-like co-
dimension 1 defect in a four-dimensional Lorentzian CFT. However, nothing we
do depends crucially on this, and everything will generalize straightforwardly
to other dimensions and signatures for both the CFT and the defect.

This EFT can also be used in the simpler case where conformal symmetry is
broken to Poincaré symmetry via a constant VEV $\langle\Phi\rangle\sim f$. We
will see that the EFT naturally organizes as an expansion in various powers of
$1/f$. For example, the two-point function $\langle\phi(x)\phi(0)\rangle$
organizes as an expansion in various powers of $1/(fx)$, which is good at long
distances (complementary to the short distance limit which can be probed by
the operator product expansion), and that the leading 1-loop quantum
correction is universal, independent of the higher derivative operators in the
EFT.

Conventions: $D$ is the spacetime dimension, and we use the mostly plus metric
signature. The curvature conventions are those of [21].

##  2 The EFT of conformal symmetry

The EFT we seek should describe the fluctuations of fields around the symmetry
breaking VEV (1.1). This is the EFT that describes the spontaneous breaking of
conformal symmetry, and which was studied many years ago as a prototype for
spontaneously broken spacetime symmetries [22, 23]. It is equivalent [24, 25,
26] to the theory of a co-dimension 1 brane fluctuating in a fixed background
anti-de Sitter space. The same EFT also plays a key role in the proof of the
$a$-theorem [27, 28], and its $S$-matrix satisfies non-trivial soft theorems
[29, 30, 31, 32].

One well-known way to construct this EFT is from the coset perspective (see
e.g. [33, 11]). However, in the following we describe an alternative and more
direct method of constructing it for arbitrary conformal weights, which will
prove useful in the rest of the paper, and which starts with fields that
linearly realize conformal symmetry.

###  2.1 Direct construction

Our goal is to construct the EFT of a weight $\Delta$ scalar conformal primary
field $\Phi$. The symmetries that must be maintained are the usual linearly-
realized conformal symmetries,

$\displaystyle\delta\Phi$ $\displaystyle=$
$\displaystyle-(x^{\mu}\partial_{\mu}+\Delta)\Phi\ ,$ (2.1)
$\displaystyle\delta_{\mu}\Phi$ $\displaystyle=$
$\displaystyle-\left(2x_{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial_{\mu}+2x_{\mu}\Delta\right)\Phi\,,$
(2.2)

which are the scale transformation and special conformal transformations,
respectively.

We construct the EFT by writing all conformally invariant terms, order by
order in powers of derivatives. We allow for terms which are non-analytic in
the fields, because we will ultimately be expanding around a conformally non-
invariant VEV111This is similar to the philosophy of the “Higgs EFT” as
opposed to the “Standard Model EFT” in the context of electroweak symmetry
breaking [34, 35]..

Scale invariance is easy to impose; it is equivalent to demanding that each
term in the Lagrangian density has total operator dimension equal to the
spacetime dimension $D$, so that there are no dimensionful couplings. We will
assume throughout that $\Delta\not=0$ and $D>2$, since other subtleties arise
otherwise.

At zeroth order in derivatives, the only scale invariant term is

${\cal L}_{0}=\Phi^{D\over\Delta}\ .$ (2.3)

This term is also invariant under special conformal transformations, so this
is our complete zeroth order Lagrangian.

At second order in derivatives, the only possible scale invariant term, up to
total derivatives, is

${\cal L}_{2}=\Phi^{{D-2\over\Delta}}{(\partial\Phi)^{2}\over\Phi^{2}}\ .$
(2.4)

This is also invariant under the special conformal transformations, so this is
our 2-derivative Lagrangian.

At fourth order in derivatives, there are three possible scale invariant
terms, up to total derivatives:

${\Phi^{{D-4\over\Delta}}{(\square\Phi)^{2}\over\Phi^{2}}},\ \ \
{\Phi^{{D-4\over\Delta}}{(\partial\Phi)^{2}\square\Phi\over\Phi^{3}}},\ \ \
{\Phi^{{D-4\over\Delta}}{(\partial\Phi)^{4}\over\Phi^{4}}}\ .$ (2.5)

However, imposing special conformal invariance, only two linear combinations
of these three terms are invariant. For later convenience we choose these
combinations to be

$\displaystyle{\cal
L}_{4}={\Phi^{{D-4\over\Delta}}}\left[{(\square\Phi)^{2}\over\Phi^{2}}-\frac{(2\Delta-D+2)(2\Delta-D+4)}{4\Delta^{2}}{(\partial\Phi)^{4}\over\Phi^{4}}\right],\
\ \ $ $\displaystyle{\cal
L}_{4}^{\prime}=\Phi^{{D-4\over\Delta}}\left[{(\partial\Phi)^{2}\square\Phi\over\Phi^{3}}-\frac{2\Delta-D+3}{2\Delta}{(\partial\Phi)^{4}\over\Phi^{4}}\right]\
.$ (2.6)

This construction can be continued to all higher orders in derivatives; at
each derivative order there will be some finite number of independent scale
invariant terms up to total derivatives, some subspace of these will be fully
conformal invariant, and a basis of this subspace forms the EFT Lagrangian at
this derivative order.

The full Lagrangian is then the sum of all these terms, with arbitrary
coefficients, organized as a derivative expansion,

${\cal L}=c_{0}{\cal L}_{0}+c_{2}{\cal L}_{2}+c_{4}{\cal
L}_{4}+c_{4}^{\prime}{\cal L}_{4}^{\prime}+\cdots\ .$ (2.7)

We will let $\\{c\\}_{4}\equiv\\{c_{4},c_{4}^{\prime}\\}$,
$\\{c\\}_{6}\equiv\\{c_{6},c_{6}^{\prime},c_{6}^{\prime\prime},\ldots\\}$,
etc. denote the sets of coefficients of terms at each derivative order.

###  2.2 Comparison to the coset construction

The usual approach to constructing this theory is the coset approach,
equivalent to the geometric method as described in [33, 11]. Here we will
extend this approach to arbitrary $\Delta$ and see that it is equivalent to
the above direct approach.

We consider the conformally flat metric

$g_{\mu\nu}=\left(\Phi^{1/\Delta}\over\Lambda\right)^{2}\eta_{\mu\nu}\ ,$
(2.8)

where $\Lambda$ is some artificially introduced energy scale that allows us to
ensure that the metric has the correct dimensions. This quantity is purely a
calculational convenience, and will cancel out at the end of our calculations.
We then write diffeomorphism invariants using this metric, with an overall
scale set by $\Lambda$. These diffeomorphism invariants are equivalent to the
basic building blocks of the coset construction. The set of all diffeomorphism
invariants at a given derivative order will in general be larger than the
number of conformally invariant Lagrangians, because some invariants may be
degenerate on the specific conformally flat metric (2.8). But no invariant
Lagrangians will be missed in this way, with one exception in even $D$ at
$D$-th order in derivatives: the Wess-Zumino term (see below.)

Starting at zeroth order in derivatives, the only diffeomorphism invariant is
the cosmological constant. When evaluated on (2.8) this reproduces the zeroth
order Lagrangian (2.3),

$\Lambda^{D}\int d^{D}x\ \sqrt{-g}=\int d^{D}x\ \Phi^{D\over\Delta}\,.$ (2.9)

At second order, the only diffeomorphism invariant is the Einstein-Hilbert
term. When evaluated on (2.8) this reproduces the second order Lagrangian
(2.4) up to total derivatives,

$\Lambda^{D-2}\int d^{D}x\ \sqrt{-g}R=\frac{(D-2)(D-1)}{\Delta^{2}}\int
d^{D}x\ \Phi^{{D-2\over\Delta}}{(\partial\Phi)^{2}\over\Phi^{2}}\,.$ (2.10)

At fourth order, there are 3 different curvature invariants up to total
derivatives: $R^{2}$, $R_{\mu\nu}^{2}$ and $R_{\mu\nu\rho\sigma}^{2}$. However
we need not consider anything made from the Riemann tensor, since the Weyl
tensor vanishes when evaluated on the conformally flat metric (2.8). Thus, we
have two possible invariants, and evaluated on (2.8) they recover, up to total
derivatives, linear combinations of the invariants (2.6),

$\displaystyle\Lambda^{D-4}\int d^{D}x\ \sqrt{-g}R^{2}=\int d^{D}x\
\frac{4(D-1)^{2}}{\Delta^{2}}{\cal
L}_{4}+\frac{4(D-1)^{2}(D-2\Delta-2)}{\Delta^{3}}{\cal L}_{4}^{\prime}\,,$
$\displaystyle\Lambda^{D-4}\int d^{D}x\ \sqrt{-g}R_{\mu\nu}^{2}=\int d^{D}x\
\frac{(D-1)D}{\Delta^{2}}{\cal
L}_{4}+\frac{(D-2)\left(3D^{2}-8D+8\right)-4(D-1)D\Delta}{2\Delta^{3}}{\cal
L}_{4}^{\prime}\,.$ (2.11)

The $2\times 2$ matrix mapping these two curvature invariants to the two
Lagrangians has full rank for $D\not=4$, and so in this case we can recover
both ${\cal L}_{4}$ and ${\cal L}_{4}^{\prime}$ from the curvatures by solving
(2.11) for ${\cal L}_{4}$, ${\cal L}_{4}^{\prime}$. For $D=4$ however, this
matrix has rank 1, and we cannot recover both Lagrangians from the curvature
invariants. In this case both curvature invariants give the same linear
combination:

$\int d^{4}x\ \sqrt{-g}R^{2}=\int d^{4}x\
\sqrt{-g}R_{\mu\nu}^{2}={12\over\Delta^{2}}\int d^{4}x\ {\cal
L}_{4}-{2(\Delta-1)\over\Delta}{\cal L}_{4}^{\prime}\,,\ \ \ D=4\,.$ (2.12)

The other linear combination is a Wess-Zumino term, which cannot be
constructed from the curvature invariants (see [33] for details on how to
construct it within the coset formalism222A quick way to do it is to form the
Gauss-Bonnet combination in general $D$,  $\Lambda^{D-4}\int
d^{D}x\sqrt{-g}\left(R_{\mu\nu\rho\sigma}^{2}-4R_{\mu\nu}^{2}+R^{2}\right)=-\frac{2(D-4)(D-3)(D-2)}{\Delta^{3}}\int
d^{D}x\ {\cal L}_{4}^{\prime}\,.$ (2.13) This results in a quantity that is
proportional to $D-4$ because in $D=4$ the Gauss-Bonnet combination becomes a
total derivative. Stripping off the $D-4$ factor yields the Wess-Zumino term
[36]. ). In any even $D$, a similar Wess-Zumino term exists at $D$-derivative
order.

##  3 Breaking to Poincaré

To start with, we demonstrate the power of this EFT technique by applying it
to simplest case, where conformal symmetry breaks to Poincaré symmetry. This
will serve as a warmup for our main case of interest, defect CFT, and
illustrates most of the general features.

To describe the breaking to Poincaré we expand around a constant VEV,

$\langle\Phi\rangle=f^{\Delta},\ \ \ \Phi=f^{\Delta}\left(1+{1\over f^{{D\over
2}-1}}\phi\right)\,,$ (3.1)

where $f$ is a constant with mass dimension 1, and the coefficient for the
fluctuation field $\phi$ is chosen so that $\phi$ will be canonically
normalized.

We now expand the Lagrangian (2.7) in powers of $\phi$. Demanding the absence
of the tadpole so that the constant VEV is a solution to the equations of
motion requires

$c_{0}=0\ .$ (3.2)

At second order in $\phi$ we then have

${\cal L}_{\phi,2}=-{1\over 2}(\partial\phi)^{2}+{c_{4}\over
f^{2}}(\square\phi)^{2}+{\cal O}\left({\partial^{6}\phi^{2}\over
f^{4}},\\{c\\}_{6}\right)+{\cal O}\left({\partial^{8}\phi^{2}\over
f^{6}},\\{c\\}_{8}\right)+\cdots\ ,$ (3.3)

where we have used the freedom to scale the field (and to flip the overall
sign of the Lagrangian if necessary) to choose $c_{1}=-{1\over 2}$ so that
$\phi$ has a canonically normalized 2-derivative kinetic term. These quadratic
terms organize as a power series in $1/f^{2}$, with coefficients of the higher
order terms coming from the higher derivative terms in (2.7). At cubic order
we have

${\cal L}_{\phi,3}={1\over f^{{D\over 2}-1}}\left[{2\Delta-D+2\over
2\Delta}\phi(\partial\phi)^{2}+{\cal O}\left({\partial^{4}\phi^{3}\over
f^{2}},\\{c\\}_{4}\right)+\cdots\right]\ .$ (3.4)

The cubic terms organize as a power series in $1/f^{2}$, starting at order
$1/f^{{D\over 2}-1}$, with coefficients of the higher order terms coming from
the higher derivative terms in (2.7). At quartic order we have

${\cal L}_{\phi,4}={1\over
f^{{D}-2}}\left[-\frac{(2\Delta-D+2)(3\Delta-D+2)}{4\Delta^{2}}\phi^{2}(\partial\phi)^{2}+{\cal
O}\left({\partial^{4}\phi^{4}\over f^{2}},\\{c\\}_{4}\right)+\cdots\right]\ .$
(3.5)

The quartic terms organize as a power series in $1/f^{2}$, starting at order
$1/f^{{D}-2}$, with coefficients of the higher orders coming from the higher
derivative terms in (2.7). This pattern continues: the terms at $n$-th order
organize in powers of $1/f^{2}$, starting at order $1/f^{(n-2)\left({D\over
2}-1\right)}$.

###  3.1 2-point function

We can use this effective theory to compute correlators systematically as a
power series expansion in $1/f$. These will depend on the free coefficients
$c_{4},c_{4}^{\prime},\cdots$, parametrizing the higher derivative terms in
the action, which also serve as counterterms to absorb divergences. As we will
see, there are also some universal parts that do not depend on these
coefficients.

To illustrate, consider the two point function $\langle\Phi(x)\Phi(0)\rangle$.
The broken conformal symmetry puts no constraint on the functional form of
this correlator beyond the usual constraints from Poincaré invariance that
tell us it must be a function of the magnitude of the invariant distance
between the two points. Expanding using (3.1) and using the assumption
$\langle\phi\rangle=0$, we have

$\langle\Phi(x)\Phi(0)\rangle=f^{2\Delta}\left[1+{1\over
f^{D-2}}\langle\phi(x)\phi(0)\rangle\right]\ .$ (3.6)

Now $\langle\phi(x)\phi(0)\rangle$ can be computed using Feynman diagrams in
the effective theory.

From power counting [37, 38] we can see that in dimensional regularization the
diagrams contributing to the momentum space 2-point function will scale as

$\langle\phi(p)\phi(-p)\rangle\sim{1\over p^{2}}\left(p\over
f\right)^{(D-2)L}\left(c\,{p\over f}\right)^{\sum_{n,k}(k-2)V_{n,k}}\,,$ (3.7)

where $V_{n,k}$ is the number of vertices with $n$ fields and $k$ derivatives
in the diagram, $L$ is the number of loops, and $c$ stands generically for the
coefficients of the higher-derivatve terms.

Suppose that we are only interested in the correlator away from $x^{2}=0$, so
that we can ignore terms analytic in $p^{2}$ (which only contribute when
$x^{2}=0$).

At tree level, $L=0$, the only corrections come from the higher-order vertices
in (3.3), and these contributions are all analytic in $p^{2}$. So all that
remains at separated points is the zeroth order propagator of the free kinetic
term,

$\langle\phi(p)\phi(-p)\rangle_{0-{\rm loop}}=-{i\over p^{2}}\ .$ (3.8)

The leading corrections at separated points come from $L=1$, with no
insertions of higher-order vertices. The diagrams are shown in Figure 1.

Figure 1: Contributions to the 2-point function at 1-loop.

The only diagram that contributes is the first one, since the rest vanish in
dimensional regularization. For $D$ even, there is a divergence that is
proportional to

$\sim{1\over\epsilon}{(2\Delta-D+2)^{2}\over\Delta^{2}}{1\over
f^{D-2}}(p^{2})^{{D\over 2}-2}\,,$ (3.9)

where $\epsilon$ is the dimensional regularization parameter. This is analytic
in $p^{2}$, is absorbed by one of the higher derivative tree-level
counterterms, and does not contribute at separated points. The finite part is
logarithmic in $p^{2}$,

$\langle\phi(p)\phi(-p)\rangle_{1-{\rm loop}}={i^{D+1}\over
2^{2D-1}\pi^{D-1\over 2}\Gamma\left({D\over
2}-1\right)}{(2\Delta-D+2)^{2}\over\Delta^{2}}{1\over f^{D-2}}(p^{2})^{{D\over
2}-2}\log(p^{2}/\mu^{2})\,,\ \ \ D\ {\rm even}\ ,$ (3.10)

where $\mu$ is the dimensional regularization scale whose value is adjusted
through the same counterterm that absorbs the divergence, and does not affect
separated points.

For odd $D$, the $L=1$ contribution (3.7) involves an odd power of $p^{2}$, so
it is not analytic. This means that there cannot be any UV divergence, because
the counterterms can only absorb analytic terms. Indeed, the loop is finite in
dimensional regularization and gives

$\langle\phi(p)\phi(-p)\rangle_{1-{\rm loop}}={i^{D}\over
2^{2D-1}\pi^{D-3\over 2}\Gamma\left({D\over
2}-1\right)}{(2\Delta-D+2)^{2}\over\Delta^{2}}{1\over f^{D-2}}(p^{2})^{{D\over
2}-2}\,,\ \ \ D\ {\rm odd}\ .$ (3.11)

Going back to position space, the 1-loop contribution gives a contribution to
the correlator at separated points that is proportional to
$\sim{1/x^{2(D-2)}}$. Including the position space version of the tree level
part (3.8), which is proportional to $\sim{1/x^{D-2}}$, and substituting back
into (3.6), we see that the full 2-point function takes the schematic form

$\displaystyle\langle\Phi(x)\Phi(0)\rangle$ $\displaystyle=$ $\displaystyle
f^{2\Delta}\left[1+{1\over f^{D-2}}\left(\langle\phi(x)\phi(0)\rangle_{0-{\rm
loop}}+\langle\phi(x)\phi(0)\rangle_{1-{\rm loop}}+\cdots\right)\right]$
(3.12) $\displaystyle\sim$ $\displaystyle f^{2\Delta}\left[1+{1\over
f^{D-2}}{1\over x^{D-2}}+{1\over f^{2(D-2)}}{1\over
x^{2(D-2)}}+\cdots\right]\,,$

where we have ignored a possible dimensionless constant in each term.

The loop expansion is an expansion in powers of ${1/(fx)^{D-2}}$, and the one-
loop contribution is independent of the higher-derivative coefficients and is
a universal prediction of the EFT. At higher orders in the expansion,
insertions of the higher derivative terms into loops will bring in further
factors of $1/(fx)^{2}$. The EFT expansion is thus valid for large $x\gg 1/f$,
which is good at long distances/low energies, as expected. Note that this is
opposite to the regime $x\ll 1/f$ for which the operator product expansion of
the underlying CFT can be used to expand the correlator in powers of $xf$
[39].

##  4 Defect CFT breaking

We now turn to the main case of interest, the breaking of conformal symmetry
to a lower dimensional conformal symmetry in the presence of a maximally
symmetric defect. We expand the EFT around the desired time-dependent symmetry
breaking VEV (1.1),

$\langle\Phi\rangle={C^{\Delta}\over{(-t)}^{\Delta}}\ ,$ (4.1)

where the dimensionless constant $C$ parametrizes the strength of the symmetry
breaking, and we take $C>0$ without loss of generality.

A general way to parametrize the fluctuations around this VEV is to write

$\Phi=\langle\Phi\rangle F(\phi)\ ,$ (4.2)

with $\phi$ the field parametrizing the fluctuations, and $F$ some arbitrary
function that is analytic at the origin and satisfies $F(0)=1$. The metric
(2.8) in term of $\phi$ is then

$g_{\mu\nu}=\left(\Phi^{1/\Delta}\over\Lambda\right)^{2}\eta_{\mu\nu}={1\over\Lambda^{2}}{C^{2}\over
t^{2}}F(\phi)^{2\Delta}\eta_{\mu\nu}=F(\phi)^{2\Delta}\bar{g}_{\mu\nu}\ ,$
(4.3)

where $\bar{g}_{\mu\nu}$ is a fake (meaning we introduce it as an unphysical
mathematically useful artifact) de Sitter metric with a Hubble scale,

$\bar{g}_{\mu\nu}\equiv{1\over H^{2}t^{2}}\eta_{\mu\nu},\ \ \
H\equiv{\Lambda\over C}\ .$ (4.4)

Thus, the effective theory of fluctuations is obtained by expanding curvature
invariants around de Sitter space, and the unbroken $\mathfrak{so}(D,1)$
symmetry of the full conformal symmetry $\mathfrak{so}(D,2)$ is manifest.

In [11], an exponential parametrization $F(\pi)=e^{\pi}$ was used. For our
purposes, it will be more straightforward to instead use the linear
parametrization $F(\phi)=1+{(-t)^{{D\over 2}-1}\over C^{{D\over 2}-1}}\,\phi$,
so that we have

$\Phi={C^{\Delta}\over{(-t)}^{\Delta}}\left(1+{(-t)^{{D\over 2}-1}\over
C^{{D\over 2}-1}}\,\phi\right)\,,$ (4.5)

where we have chosen the factor in front of the fluctuation $\phi$ to ensure
that this field will be canonically normalized when we expand the action.

We now expand the Lagrangian (2.7) in powers of $\phi$ using (4.5). At linear
order there is a tadpole term. Demanding that this vanishes up to total
derivatives, so that the background (4.1) is a solution to the equations of
motion, gives the condition333We will show later that the one-loop correction
does not contribute to the tadpole term.

$c_{0}+{1\over C^{2}}c_{2}\Delta^{2}+{1\over C^{4}}\left[c_{4}{\Delta^{2}\over
4}\left(3D^{2}-10D+4-4\Delta(D-1)\right)-c_{4}^{\prime}{\Delta^{3}(D-1)\over
2}\right]+{\cal O}\left({1\over C^{6}},\\{c\\}_{6}\right)+\cdots=0\,.$ (4.6)

This condition is an expansion in powers of $1/C^{2}$, with coefficients
arising from the Lagrangian terms with higher derivatives. We use this
condition to solve for $c_{0}$, keeping only up to the order in $1/C^{2}$ that
we are ultimately interested in. In addition, we will use the freedom to
rescale the field $\Phi$ (and to flip the overall sign of the Lagrangian if
necessary) to choose

$c_{2}=-{1\over 2}\,,$ (4.7)

which will ensure that the kinetic term for the fluctuation $\phi$, as defined
in (4.5), is canonically normalized.

At higher orders we get the following terms up to total derivatives: at second
order in $\phi$ we have

$\displaystyle{\cal L}_{\phi,2}$ $\displaystyle=$ $\displaystyle{1\over
2}\dot{\phi}^{2}-{1\over
2}(\vec{\nabla}\phi)^{2}+\frac{D(D+2)}{8t^{2}}\phi^{2}$ (4.8)
$\displaystyle+{1\over
C^{2}}\bigg{[}c_{4}\bigg{(}t^{2}\ddot{\phi}^{2}-2t^{2}(\vec{\nabla}\dot{\phi})^{2}+(2\Delta(D-1)-D^{2}+5D-2)\dot{\phi}^{2}+t^{2}(\vec{\nabla}^{2}\phi)^{2}$
$\displaystyle-\left(2(D-1)\Delta-(D-1)(D-4)\right)(\vec{\nabla}\phi)^{2}+\frac{8D(D-1)(D+2)\Delta-{D(D+2)(5D^{2}-18D+8)}}{16t^{2}}\phi^{2}\bigg{)}$
$\displaystyle+c_{4}^{\prime}\Delta(D-1)\bigg{(}\dot{\phi}^{2}-(\vec{\nabla}\phi)^{2}+\frac{D(D+2)}{4t^{2}}\phi^{2}\bigg{)}\bigg{]}$
$\displaystyle+{\cal O}\left({1\over C^{4}},\\{c\\}_{6}\right)+\cdots\ .$

These quadratic terms organize as a power series in $1/C^{2}$, with
coefficients coming from the higher order terms in (2.7).

At cubic order in $\phi$ we have

$\displaystyle{\cal L}_{\phi,3}={1\over C^{{D\over 2}-1}}{(-t)^{{D\over
2}-1}\over 2\Delta}\bigg{[}$
$\displaystyle(2\Delta-D+2)\left(\phi\dot{\phi}^{2}-\phi(\vec{\nabla}\phi)^{2}\right)+{2\left(D^{2}+6D-4\right)\Delta-D^{3}-6D^{2}+12D-8\over
12t^{2}}\phi^{3}$ (4.9) $\displaystyle+{\cal O}\left({1\over
C^{2}},\\{c\\}_{4}\right)+\cdots\bigg{]}\,.$

These terms also organize as a power series in $1/C^{2}$, starting at order
${1/C^{{D\over 2}-1}}$.

At quartic order in $\phi$ we have

$\displaystyle{\cal L}_{\phi,4}={1\over C^{D-2}}{(-t)^{D-2}\over
4\Delta^{2}}\bigg{[}$
$\displaystyle(6\Delta^{2}-5\Delta(D-2)+(D-2)^{2})\left(\phi^{2}\dot{\phi}^{2}-\phi^{2}(\vec{\nabla}\phi)^{2}\right)$
(4.10)
$\displaystyle+{4(10D-9)\Delta^{2}-3\left(13D^{2}-28D+20\right)\Delta+9D^{3}-30D^{2}+44D-24\over
12t^{2}}\phi^{4}$ $\displaystyle+{\cal O}\left({1\over
C^{2}},\\{c\\}_{4}\right)+\cdots\bigg{]}\,.$

These terms again organize as a power series in $1/C^{2}$, starting at order
${1/C^{D-2}}$. This pattern continues: the terms at $n$-th order in the
fluctuation organize in powers of $1/C^{2}$, starting at order
$1/C^{(n-2)\left({D\over 2}-1\right)}$.

##  5 Defect CFT 2-point function

We can now use the EFT to compute correlators of $\Phi$ in the broken vacuum
(4.1) as an expansion in powers of $1/C$. Consider for example the 2-point
function
$\langle\Phi(t,\mathbf{x})\Phi(t^{\prime},\mathbf{x}^{\prime})\rangle$. The
broken conformal symmetry dictates that the general form of this correlator is
an arbitrary function $F$ of the cross ratio $\xi$ [40, 41, 42, 43, 44, 2, 45,
46, 20],

$\langle\Phi(t,\mathbf{x})\Phi(t^{\prime},\mathbf{x}^{\prime})\rangle={F(\xi)\over(-t)^{\Delta}(-t^{\prime})^{\Delta}},\
\ \ \xi\equiv{-(t-t^{\prime})^{2}+(\mathbf{x}-\mathbf{x}^{\prime})^{2}\over
4(-t)(-t^{\prime})}\,.$ (5.1)

Substituting in the expansion (4.5) and using that $\langle\phi\rangle=0$, we
have

$\langle\Phi(t,\mathbf{x})\Phi(t^{\prime},\mathbf{x}^{\prime})\rangle={C^{2\Delta}\over(-t)^{\Delta}(-t^{\prime})^{\Delta}}\left[1+{1\over
C^{D-2}}{(-t)^{{D\over 2}-1}(-t^{\prime})^{{D\over
2}-1}}\langle\phi(t,\mathbf{x})\phi(t^{\prime},\mathbf{x}^{\prime})\rangle\right]\,.$
(5.2)

We can then compute the fluctuation correlator
$\langle\phi(t,\mathbf{x})\phi(t^{\prime},\mathbf{x}^{\prime})\rangle$ from
the expanded Lagrangian, which will take the form of a loop expansion in
powers of $1/C^{D-2}$ and higher derivative corrections of order $1/C^{2}$.

We now turn to computing some of the leading corrections. The tree level
correlator is easily evaluated. The next to leading order correction is more
difficult and we will restrict ourselves to computing the equal time, late
time limit, $t=t^{\prime}\to 0$.

###  5.1 Leading part

We compute the leading $C$-independent part of the fluctuation correlator
$\langle\phi(t,\mathbf{x})\phi(t^{\prime},\mathbf{x}^{\prime})\rangle$ from
the $C$ independent part of the quadratic action; the first line of (4.8).
This has the same form as that of a massive scalar on de-Sitter space with
mass to Hubble ratio $m^{2}/H^{2}=-D$.444Note that this is the mass value for
a $k=1$ Galileon/DBI shift-symmetric scalar in de Sitter [47, 48, 49, 50, 51].
The shift symmetry is the broken conformal symmetry.

Expanding the field in mode functions

$\phi(t,\mathbf{x})={1\over(2\pi)^{(D-1)/2}}\int d^{D-1}\mathbf{k}\left[\
a_{\mathbf{k}}f_{k}(t)e^{i\mathbf{k}\cdot\mathbf{x}}+a_{\mathbf{k}}^{\dagger}f_{k}^{\ast}(t)e^{-i\mathbf{k}\cdot\mathbf{x}}\right]\
,$ (5.3)

where the mode functions should satisfy the equation

$f_{k}(t)^{\prime\prime}+\left[k^{2}-{D(D+2)\over 4t^{2}}\right]f_{k}(t)=0\ ,$
(5.4)

which depends only on the magnitude $k\equiv|\mathbf{k}|$ of the spatial
momentum. We will assume the standard Bunch-Davies boundary conditions

$f_{k}(t)\underset{t\rightarrow-\infty}{\longrightarrow}{1\over\sqrt{2k}}e^{-ikt}\
.$ (5.5)

which gives the solution

$f_{k}(t)={\sqrt{\pi}\over 2}\sqrt{-t}\,e^{i\pi\left(D+2\over
4\right)}H_{D+1\over 2}^{(1)}\left(-kt\right)\ .$ (5.6)

Using the usual creation/annihilation operator relations,
$[a_{\mathbf{k}},a^{\dagger}_{\mathbf{k}^{\prime}}]=\delta^{D-1}(\mathbf{k}-\mathbf{k}^{\prime})$,
$[a_{\mathbf{k}},a_{\mathbf{k}^{\prime}}]=[a_{\mathbf{k}}^{\dagger},a^{\dagger}_{\mathbf{k}^{\prime}}]=0$,
$a_{\mathbf{k}}|0\rangle=\langle 0|a_{\mathbf{k}}^{\dagger}=0$, we can now
compute the spatial Fourier transform of the 2-point function,

$\displaystyle\langle\phi(t,\mathbf{k})\phi(t^{\prime},\mathbf{k}^{\prime})\rangle=\int
d^{D-1}\mathbf{x}d^{D-1}\mathbf{x}^{\prime}e^{i\mathbf{k}\cdot\mathbf{x}}e^{i\mathbf{k}^{\prime}\cdot\mathbf{x}^{\prime}}\langle\phi(t,\mathbf{x})\phi(t^{\prime},\mathbf{x}^{\prime})\rangle=(2\pi)^{D-1}\delta^{D-1}(\mathbf{k}+\mathbf{k}^{\prime})f_{k}(t)f_{k^{\prime}}^{\ast}(t^{\prime})\
.$

###  5.2 Next to leading order corrections

We now compute the 2-point correlation function beyond leading order. We will
specialize to the case $D=4$, but will keep $D$ explicit in the following
expressions in order to make dimensional regularization easier to implement.
The contributions to this part are the tree level diagrams with higher
derivative terms from the quadratic action in (4.8), and the loop corrections
from the cubic interaction and quartic interaction as shown in figure 1.

We will make use of the Mellin-Barnes representation [52] to compute the
corrections via the in-in formalism and to regulate the loop [53]. The mode
function (5.6) expressed in the Mellin-Barnes representation has the following
form,

$f_{k}(t)=\frac{e^{i\pi\nu-\frac{i\pi}{4}}}{\sqrt{4\pi}}\int_{c-i\infty}^{c+i\infty}\frac{ds}{2\pi
i}\Gamma\left(s+\frac{\nu}{2}\right)\Gamma\left(s-\frac{\nu}{2}\right)\left(-\frac{i}{2}k\right)^{-2s}(-t)^{-2s+\frac{1}{2}}\,,$
(5.8)

where $\nu\equiv\frac{D+1}{2}$, and the contour is chosen to be a straight
line parallel to the imaginary axis, intersecting the real axis at $c$, and is
closed to the left. We require $c>\left|\mathfrak{Re}\frac{\nu}{2}\right|$ to
include all the poles within the contour. In flat spacetime, the Fourier
transform provides a convenient representation for mode functions since its
variable $k^{0}$ is the energy, and the resulting $\delta$-functions that
arise during calculations enforce conservation of energy because of time-
translations symmetry. Similarly, in de Sitter space, the Mellin-Barnes
representation is convenient because the variable $s$ can be thought of as the
eigenvalue of the dilatation symmetry, with the corresponding
$\delta$-functions enforcing its conservation.

The equal time in-in 2-point function evaluated at time $t_{0}$ is given by

$B_{2}\equiv\lim_{t\to-\infty}\braket{0}{~{}\mathcal{U}^{\dagger}(t_{0},t)\phi(t_{0},\mathbf{k})\phi(t_{0},\mathbf{k}^{\prime})\mathcal{U}(t_{0},t)}{0}\,,$
(5.9)

where $\mathcal{U}$ is the full time evolution operator, given in terms of the
interaction picture Hamiltonian $\mathcal{H}_{\text{int}}(t)$ via,

$\mathcal{U}=1-i\int_{-\infty}^{t_{0}}dt\mathcal{H}_{\text{int}}(t)-\frac{1}{2}\int_{-\infty}^{t_{0}}dt_{1}\int_{-\infty}^{t_{0}}dt_{2}\mathcal{T}\left\\{\mathcal{H}_{\text{int}}(t_{1})\mathcal{H}_{\text{int}}(t_{2})\right\\}+\cdots\,,$
(5.10)

where $\mathcal{T}\\{\cdots\\}$ represents time-ordering.

Define the bulk-to-boundary propagator for a time-ordered vertex as:

$G_{B+}(k,t)=f_{k}(t)f^{*}_{k}(t_{0})\,,$ (5.11)

and the bulk-to-bulk propagator between two time-ordered vertices as,

$G_{++}(k,t_{1},t_{2})=\theta(t_{1}-t_{2})f^{*}_{k}(t_{1})f_{k}(t_{2})+\theta(t_{2}-t_{1})f^{*}_{k}(t_{2})f_{k}(t_{1})\,.$
(5.12)

The propagators for anti-time ordered vertices can then be obtained through
conjugation,

$G_{B-}(k,t)=G_{B+}(k,t)^{*}=f^{*}_{k}(t)f_{k}(t_{0})\,,$ (5.13)
$G_{--}(k,t_{1},t_{2})=G_{++}(k,t_{1},t_{2})^{*}=\theta(t_{1}-t_{2})f^{*}_{k}(t_{2})f_{k}(t_{1})+\theta(t_{2}-t_{1})f^{*}_{k}(t_{1})f_{k}(t_{2})\,.$
(5.14)

The bulk-to-bulk propagators connecting a time-ordered vertex at time $t_{1}$
and an anti-time ordered vertex at time $t_{2}$ are given by

$G_{+-}(k,t_{1},t_{2})=f^{*}_{k}(t_{2})f_{k}(t_{1})~{},\quad
G_{-+}(k,t_{1},t_{2})=G_{+-}(k,t_{2},t_{1})=f_{k}(t_{2})f^{*}_{k}(t_{1})\,.$
(5.15)

With these ingredients at hand, we can start computing the $1/C^{2}$
corrections from the various diagrams. We will see that, analogously to the
Poincaré case in Section 3, only the diagram on the left in Figure 1
contributes non-trivially.

####  5.2.1 Higher derivative term

We start with the tree level correction with a single vertex drawn from the
terms with power $1/C^{2}$ in 4.8. Once the equations of motion for the
external lines are taken into account, the term proportional to
$c_{4^{\prime}}$ reduces to a boundary term,

$-i\frac{c_{4^{\prime}}\Delta(D-1)}{C^{2}}f_{k}(t_{0})\dot{f}_{k}(t_{0})f_{k}^{\ast}(t_{0})f_{k}^{\ast}(t_{0})\,.$
(5.16)

After taking the late time limit and summing with the anti-time ordered
vertex, this boundary term contributes as

$-\frac{3c_{4^{\prime}}\Delta}{2C^{2}}\frac{1}{(k^{5}t_{0}^{4})}\left(9+3(-kt_{0})^{2}+(-kt_{0})^{4}\right)=-\frac{3c_{4^{\prime}}\Delta}{C^{2}}f_{k}(t_{0})f_{k}^{\ast}(t_{0})\,,$
(5.17)

where we have explicitly set $D=4$. In fact, it is true for any even dimension
$D$ that
$\text{Re}\left(-if_{k}(t_{0})\dot{f}_{k}(t_{0})f_{k}^{\ast}(t_{0})^{2}\right)=-\frac{1}{2}f_{k}(t_{0})f_{k}^{\ast}(t_{0})$
in the late time limit. In the same spirit, we can integrate by parts in the
$c_{4}$ term, organizing the action as

$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\int\frac{dtd^{D-1}\mathbf{k}}{(2\pi)^{D-1}}\left[\left(2\Delta(D-1)-D^{2}+5D-2\right)\left(\dot{\phi}^{2}-k^{2}\phi^{2}+\frac{D(D+2)}{4t^{2}}\phi^{2}\right)\right.$
(5.18) $\displaystyle+$
$\displaystyle\left.2k^{2}\phi^{2}-\frac{D^{2}(D+2)^{2}}{16t^{2}}\phi^{2}+t^{2}\ddot{\phi}^{2}-2t^{2}k^{2}\dot{\phi}^{2}+t^{2}k^{4}\phi^{2}\right]\,.$

Since the mode functions satisfy the Hankel equation
$\ddot{f}_{k}(t)=-\frac{4k^{2}t^{2}-D(D+2)}{4t^{2}}f_{k}(t)$, we can simplify
the second line to yield

$\displaystyle-2t^{2}k^{2}\dot{\phi}^{2}+2t^{2}k^{4}\phi^{2}-\frac{1}{2}k^{2}D(D+2)\phi^{2}+2k^{2}\phi^{2}\
,$ (5.19)

which is also a pure boundary term
$\partial_{t}\left(-2t^{2}k^{2}\dot{\phi}\phi+2k^{2}t\phi^{2}\right)$. The
extra factor of $k^{2}t^{2}$ makes the boundary terms higher order in the
momentum $k$, in comparison with (5.17),

$\displaystyle-i\frac{c_{4}}{C^{2}}\left(\left[(2\Delta(D-1)-D^{2}+5D-2)-2t_{0}^{2}k^{2}\right]f_{k}(t_{0})\dot{f}_{k}(t_{0})f_{k}^{\ast}(t_{0})f_{k}^{\ast}(t_{0})+2t_{0}k^{2}f_{k}(t_{0})f_{k}(t_{0})f_{k}^{\ast}(t_{0})f_{k}^{\ast}(t_{0})\right)$
$\displaystyle\xrightarrow{\lim_{t_{0}\to
0}}-i\frac{c_{4}}{C^{2}}(2\Delta(D-1)-D^{2}+5D-2)f_{k}(t_{0})\dot{f}_{k}(t_{0})f_{k}^{\ast}(t_{0})^{2}\,.$
(5.20)

####  5.2.2 Quartic loop

We next turn to computing the correction at $\mathcal{O}(1/C^{2})$ from the
loop with a single quartic vertex, as shown in part b) of figure 1.

After integrations by parts and use of the lowest order equations of motion on
the external lines, the quartic lagrangian (4.10) takes the form

$S_{4}\propto\int
dt\frac{d^{D-1}\mathbf{k}}{(2\pi)^{D-1}}(-t)^{D-4}\phi^{4}\,,$ (5.21)

and the contribution of this to the 2-point correlation function takes the
corresponding form

$\displaystyle B_{2q}^{1+}=-\frac{i}{2}e^{\frac{\pi
i(2D+1)}{2}}f^{\ast}_{k}(t_{0})^{2}\int_{-\infty}^{t_{0}}f_{k}(t)^{2}\int\frac{d^{D-1}\mathbf{p}}{(2\pi)^{D-1}}f_{p}(t)f_{p}^{\ast}(t)\,.$
(5.22)

Inserting the Mellin-Barnes representation of the mode function (5.6), we have

$\displaystyle-\frac{ie^{\frac{\pi
i(2D+1)}{2}}}{2}f^{\ast}_{k}(t_{0})^{2}\int_{-\infty}^{t_{0}}dt\int\frac{d^{D-1}\mathbf{p}}{(2\pi)^{D-1}}\int[ds_{i}]\Gamma\left(s_{i}\pm\frac{\nu}{2}\right)\left(\frac{k}{2}\right)^{-2s_{1}-2s_{2}}\left(\frac{p}{2}\right)^{-2s_{3}-2s_{4}}$
$\displaystyle\times e^{\pi i(s_{1}+s_{2}+s_{3}-s_{4})}(-t)^{D-2-2\sum
s_{i}}\,,$ (5.23)

where $i=1,2,3,4$. The loop momentum integration is

$\int\frac{d^{D-1}\mathbf{p}}{(2\pi)^{D-1}}\frac{1}{p^{2s_{3}+2s_{4}}}=\frac{2\pi^{(D-1)/2}}{(2\pi)^{D-1}\Gamma\left(\frac{D-1}{2}\right)}\int_{0}^{\infty}\frac{dp}{p^{2s_{3}+2s_{4}-D+2}}=\frac{2\pi^{(D-1)/2}}{(2\pi)^{D-1}\Gamma\left(\frac{D-1}{2}\right)}2\pi
i\delta\left(2s_{3}+2s_{4}-D+1\right)\,,$ (5.24)

where, to ensure that the integral is convergent, we require
$Re(-2s_{3}-2s_{4}+D-1)<0$.

In this diagram, the integration over time and the loop integration are
completely separated, and each one contributes a $\delta$-function. The time
integration is

$\int_{-\infty}^{t_{0}}(-t)^{D-2-\sum s_{i}}\xrightarrow{\lim_{t_{0}\to
0}}2\pi i\delta\left(D-1-2\sum s_{i}\right)\,.$ (5.25)

If we were to directly substitute this into the correlation function, and
integrate $s_{1}$ and $s_{3}$ using the two delta functions, we would find
that the left poles and right poles overlap in the $s_{2}$ integration. We
therefore need to introduce a regulating parameter for the time integration as
follows,

$(-t_{0})^{-2\epsilon}\int_{-\infty}^{t_{0}}(-t)^{D-2-\sum
s_{i}+2\epsilon}\xrightarrow{\lim_{t_{0}\to 0}}2\pi
i\delta\left(D-1+2\epsilon-2\sum s_{i}\right)\,,$ (5.26)

where $(-t_{0})^{-2\epsilon}$ is introduced to keep the dimension correct. The
resulting quartic loop contribution to the $2$-point correlation function is
then

$\displaystyle-\frac{i}{2}e^{\frac{\pi
i(2D+1)}{2}}f^{\ast}_{k}(t_{0})^{2}(-t_{0})^{-2\epsilon}\int[ds_{i}]\Gamma\left(s_{i}\pm\frac{\nu}{2}\right)\left(\frac{k}{2}\right)^{-2s_{1}-2s_{2}}e^{\pi
i(s_{1}+s_{2}+s_{3}-s_{4})}$ (5.27) $\displaystyle\times$
$\displaystyle\frac{2\pi^{(D-1)/2}}{(2\pi)^{D-1}\Gamma\left(\frac{D-1}{2}\right)}\frac{2\pi
i\delta\left(2s_{3}+2s_{4}-D+1\right)}{2^{-2s_{3}-2s_{4}}}2\pi
i\delta\left(D-1+2\epsilon-2\sum s_{i}\right)\,.$

Since we have two delta functions here, we can choose to carry out the $s_{1}$
and $s_{3}$ integrations. We also deform the space-time dimension
$D=4+2\epsilon$ to avoid the overlap of poles in the $s_{4}$ integration, and
obtain

$\displaystyle B_{2q}^{1+}$ $\displaystyle=$ $\displaystyle\frac{e^{2\pi
i\epsilon}}{8}f^{\ast}_{k}(t_{0})^{2}\left(-\frac{kt_{0}}{2}\right)^{-2\epsilon}\frac{2\pi^{(D-1)/2}}{(2\pi)^{D-1}\Gamma\left(\frac{D-1}{2}\right)}$
(5.28) $\displaystyle\times$
$\displaystyle\int[ds_{i}]\Gamma\left(s_{2,4}\pm\frac{\nu}{2}\right)\Gamma\left(\epsilon-
s_{2}\pm\frac{\nu}{2}\right)\Gamma\left(\frac{3}{2}+\epsilon-
s_{4}\pm\frac{\nu}{2}\right)e^{\pi i(\epsilon+\frac{3}{2}-2s_{4})}\,.$

Notice that, using Barnes’ lemma, the $s_{2}$ integration yields a
$1/\epsilon$-like divergence,

$\int\frac{ds_{2}}{2\pi
i}\Gamma\left(s_{2}+\frac{\nu}{2}\right)\Gamma\left(s_{2}-\frac{\nu}{2}\right)\Gamma\left(\epsilon-
s_{2}+\frac{\nu}{2}\right)\Gamma\left(\epsilon-
s_{2}-\frac{\nu}{2}\right)=\frac{\Gamma\left(\epsilon+\nu\right)\Gamma\left(\epsilon-\nu\right)\Gamma\left(\epsilon\right)^{2}}{\Gamma\left(2\epsilon\right)}\
.$ (5.29)

The $s_{4}$ integration has an extra phase factor, which can not be computed
using Barnes’ lemma directly. Instead, we close the contour to the left, and
enclose two series of poles at $s_{2}=-n\pm\frac{\nu}{2}$. Summing over the
residues at these two series of poles, we obtain two gaussian hypergeometric
functions,

$\displaystyle\int\frac{ds_{4}}{2\pi
i}\Gamma\left(s_{4}+\frac{\nu}{2}\right)\Gamma\left(s_{4}-\frac{\nu}{2}\right)\Gamma\left(\frac{3}{2}+\epsilon-
s_{4}+\frac{\nu}{2}\right)\Gamma\left(\frac{3}{2}+\epsilon-
s_{4}-\frac{\nu}{2}\right)e^{\pi i\left(\frac{3}{2}+\epsilon-2s_{4}\right)}$
$\displaystyle=$ $\displaystyle-e^{\pi
i\left(\frac{3}{2}+\epsilon\right)}\Gamma\left(\frac{3}{2}+\epsilon\right)\left(\frac{\Gamma\left(\nu\right)\Gamma\left(1-\nu\right)}{\Gamma\left(1+\nu\right)}\Gamma\left(\frac{3}{2}+\nu+\epsilon\right)~{}_{2}F_{1}\left(\frac{3}{2}+\epsilon,\frac{3}{2}+\epsilon+\nu;1+\nu;1\right)e^{\pi
i\nu}+(\nu\to-\nu)\right)\,.$

Now, by Gauss’ theorem, when $\mathfrak{Re}(c-a-b)>0$, the hypergeometric
function becomes

$~{}_{2}F_{1}\left(a,b,c;1\right)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\
.$ (5.31)

We use this to analytically evaluate the integration in the region where
$\epsilon$ goes to zero, and the integration (5.2.2) then becomes

$\displaystyle
i\pi\frac{\Gamma\left(-2-2\epsilon\right)}{\Gamma\left(-\frac{1}{2}-\epsilon\right)}\left(\frac{\Gamma\left(-1+\epsilon\right)}{\Gamma\left(-3-\epsilon\right)}+\frac{\Gamma\left(4+\epsilon\right)}{\Gamma\left(2-2\epsilon\right)}\right)=0\,,$
(5.32)

and the quartic loop does not contribute to the 2-point correlator.

###  5.3 Two-point reducible diagram

We next compute the reducible diagram, part c) in figure 1. This diagram
contains the same massless loop as in the quartic loop case. Without
computation, we know that this loop integration gives a delta function as in
(5.24), which separates out from the other parts of the computation, and will
end up as in (5.2.2). We therefore conclude that this massless tadpole loop
diagram does not contribute to the UV divergence when computed in dimensional
regularization, in the same way as it does not contribute to flat space
amplitudes. This also explains why the there is no quantum correction to the
one-point function tadpole.

###  5.4 Cubic loop

Finally we turn to the contribution from part a) of figure 1, the irreducible
diagram with 2 cubic vertices. After integrations by parts and using the
external lowest order equations of motion, the $1/C^{2}$ cubic vertex (4.9)
reduces to the following form,

$S_{3}\sim\int
dt\frac{d^{D-1}\mathbf{k}}{(2\pi)^{D-1}}(-t)^{\frac{D}{2}-3}\phi^{3}\,.$
(5.33)

The terms contributing to the correlator coming from the expansion of (5.10),
which we call $B_{2c}^{(1)++},B_{2c}^{(1)--},B_{2c}^{(1)-+}$, in which the
subscript $c$ indicates that the correction is from cubic loop term, are
defined as follows (Notice an extra factor of $1/2$ from the Taylor expansions
of $B_{2c}^{(1)++}$ and $B_{2c}^{(1)--}$):

$\displaystyle B_{2c}^{(1)++}$ $\displaystyle\equiv$
$\displaystyle\left\langle
0\left|\hat{\phi}\left(k,t_{0}\right)\hat{\phi}\left(k,t_{0}\right)\left(-\frac{1}{2}\int_{-\infty}^{t_{0}}dt_{1}\int_{-\infty}^{t_{0}}dt_{2}\mathcal{H}_{\text{int
}}\left(t_{1}\right)\mathcal{H}_{\text{int
}}\left(t_{2}\right)\right)\right|0\right\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2}f_{k}^{\ast}(t_{0})^{2}\int_{-\infty}^{t_{0}}dt_{1,2}(-t_{1})^{\frac{D}{2}-3}(-t_{2})^{\frac{D}{2}-3}f_{k}(t_{1})f_{k}(t_{2})\int_{\mathbf{p}_{1,2}}G_{++}(p_{1},t_{1},t_{2})G_{++}(p_{2},t_{1},t_{2})\,,$

where $\int_{\mathbf{p}_{1,2}}$ is short for
$\int\frac{d^{D-1}\mathbf{p}_{1,2}}{(2\pi)^{2D-2}}\delta(\mathbf{p}_{1}+\mathbf{p}_{2}+\mathbf{k})$.
The corresponding term $B_{2c}^{--}$ is then given by the complex conjugation
of $B_{2c}^{(1)++}$. We also have

$\displaystyle B_{2c}^{(1)-+}$ $\displaystyle\equiv$
$\displaystyle\left\langle
0\left|\left((+i)\int_{-\infty}^{t_{0}}dt_{2}\mathcal{H}_{\text{int
}}\left(t_{2}\right)\right)\hat{\phi}\left(k,t_{0}\right)\hat{\phi}\left(k,t_{0}\right)\left((-i)\int_{-\infty}^{t_{0}}dt_{1}\mathcal{H}_{\text{int
}}\left(t_{1}\right)\right)\right|0\right\rangle$ $\displaystyle=$
$\displaystyle
f_{k}(t_{0})f_{k}^{\ast}(t_{0})\int_{-\infty}^{t_{0}}dt_{1,2}(-t_{1})^{\frac{D}{2}-3}(-t_{2})^{\frac{D}{2}-3}f_{k}(t_{1})f_{k}^{\ast}(t_{2})\int_{\mathbf{p}_{1,2}}G_{+-}(p_{1},t_{1},t_{2})G_{+-}(p_{2},t_{1},t_{2})\,.$

We start with $B_{2c}^{(1)++}$. Expanding the bulk-to-bulk propagator, we have

$G_{++}\left(p_{1},t_{1},t_{2}\right)G_{++}\left(p_{2},t_{1},t_{2}\right)=\theta(t_{1}-t_{2})f_{p_{1}}^{\ast}(t_{1})f_{p_{2}}^{\ast}(t_{1})f_{p_{1}}(t_{2})f_{p_{2}}(t_{2})+(t_{1}\leftrightarrow
t_{2})\,.$ (5.34)

For the two different time orderings, we denote the one containing
$\theta(t_{1}-t_{2})$ by $B_{2c}^{(1)\pm\pm,>}$ and the one containing
$\theta(t_{2}-t_{1})$ by $B_{2c}^{(1)\pm\pm,<}$.

Using the Mellin-Barnes representation for the mode functions (5.8), we have

$\displaystyle B_{2c}^{(1)++,>}=-\frac{e^{\frac{\pi
i(2D+1)}{2}}f_{k}^{\ast}(t_{0})^{2}}{(4\pi)^{3}}\int_{-\infty}^{t_{0}}dt_{1}\int_{-\infty}^{t_{1}}dt_{2}\int[ds_{i}]\Gamma\left(s_{i}+\frac{\nu_{i}}{2}\right)\Gamma\left(s_{i}-\frac{\nu_{i}}{2}\right)e^{\pi
i(s_{1}+s_{2}+s_{5}+s_{6}-s_{3}-s_{4})}$
$\displaystyle\left(\frac{k}{2}\right)^{-2(s_{1}+s_{2})}\int\frac{d^{D-1}\mathbf{p}_{1}}{(2\pi)^{D-1}}(-t_{1})^{-2(s_{1}+s_{3}+s_{4})+\frac{D-3}{2}}(-t_{2})^{-2(s_{2}+s_{5}+s_{6})+\frac{D-3}{2}}\left(\frac{p_{1}}{2}\right)^{-2(s_{3}+s_{5})}\left(\frac{|p_{1}+k|}{2}\right)^{-2(s_{4}+s_{6})}\
,$ (5.35)

and

$\displaystyle B_{2c}^{(1)++,<}=-\frac{e^{\frac{\pi
i(2D+1)}{2}}f_{k}^{\ast}(t_{0})^{2}}{(4\pi)^{3}}\int_{-\infty}^{t_{0}}dt_{1}\int_{t_{1}}^{t_{0}}dt_{2}\int[ds_{i}]\Gamma\left(s_{i}+\frac{\nu_{i}}{2}\right)\Gamma\left(s_{i}-\frac{\nu_{i}}{2}\right)e^{\pi
i(s_{1}+s_{2}+s_{3}+s_{4}-s_{5}-s_{6})}$
$\displaystyle\left(\frac{k}{2}\right)^{-2(s_{1}+s_{2})}\int\frac{d^{D-1}\mathbf{p}_{1}}{(2\pi)^{D-1}}(-t_{1})^{-2(s_{1}+s_{3}+s_{4})+\frac{D-3}{2}}(-t_{2})^{-2(s_{2}+s_{5}+s_{6})+\frac{D-3}{2}}\left(\frac{p_{1}}{2}\right)^{-2(s_{3}+s_{5})}\left(\frac{|p_{1}+k|}{2}\right)^{-2(s_{4}+s_{6})}\,,$

where $\int[ds_{i}]\equiv\Pi_{i=1}^{6}\int_{c-i\infty}^{c+i\infty}ds_{i}/2\pi
i$, and we have used momentum conservation to write
$\mathbf{p}_{2}=\mathbf{p}_{1}+\mathbf{k}$.

Note that both of the above two expressions have the same momentum integrals.
We now use the Schwinger parameterization:

$\displaystyle\int\frac{d^{D-1}\mathbf{p}}{(2\pi)^{D-1}}\frac{1}{|p|^{2s_{3}+2s_{5}}|p+k|^{2s_{4}+2s_{6}}}$
$\displaystyle=$
$\displaystyle\frac{k^{D-1-2(s_{3}-s_{4}-s_{5}-s_{6})}\Gamma\left(s_{3}+s_{4}+s_{5}+s_{6}-\frac{D-1}{2}\right)\Gamma\left(\frac{D-1}{2}-s_{3}-s_{5}\right)\Gamma\left(\frac{D-1}{2}-s_{4}-s_{6}\right)}{(4\pi)^{\frac{D-1}{2}}\Gamma\left(s_{3}+s_{5}\right)\Gamma\left(s_{4}+s_{6}\right)\Gamma\left(D-1-(s_{3}+s_{4}+s_{5}+s_{6})\right)}\
$

to perform these integrals. This standard result introduces a degeneracy in
the poles of the integrand; when evaluating the residue at the pole at
$s_{3}+s_{5}=\frac{D-1}{2}$, due to the term
$\Gamma\left(\frac{D-1}{2}-s_{3}-s_{5}\right)$, we find a cancellation between
the term $\Gamma\left(s_{3}+\cdots s_{6}-\frac{D-1}{2}\right)$ in the
numerator, and the term $\Gamma(s_{4}+s_{6})$ in the denominator. Therefore,
when shifting the contour, it is possible that we might miss some of the
necessary poles because the order in which we choose to shift the contour
depends on the arbitrary ordering of the variables. To take account of this
possibility, following [53], we introduce a new parameter that allows us to
ensure that the pole structure is manifest at all times.

Treating the momentum integral as a convolution of two copies of
$\tilde{f}_{s}(p)\equiv 1/p^{s}$, for different values of $s$, we have

$\displaystyle\int\frac{d^{D-1}\mathbf{p}}{(2\pi)^{D-1}}\frac{1}{|p|^{2s_{3}+2s_{5}}|p+k|^{2s_{4}+2s_{6}}}=\int\frac{d^{D-1}\mathbf{p}}{(2\pi)^{D-1}}\tilde{f}_{2s_{3}+2s_{5}}(\mathbf{p})\tilde{f}_{2s_{4}+2s_{6}}(\mathbf{p}+\mathbf{k})$
$\displaystyle=\int d^{D-1}\mathbf{y}\
e^{-i\mathbf{k}\cdot\mathbf{y}}f_{2s_{3}+2s_{5}}(\mathbf{y})f_{2s_{4}+2s_{6}}(\mathbf{y})\
,$ (5.38)

where
$f_{s}(\mathbf{y})=\int\frac{d^{D-1}\mathbf{p}}{(2\pi)^{D-1}}e^{i\mathbf{p}\cdot\mathbf{y}}\tilde{f}_{s}(p)=\frac{1}{2^{s}\pi^{(D-1)/2}}\frac{\Gamma\left(\frac{D-s}{2}\right)}{\Gamma\left(\frac{s}{2}\right)}y^{s-D}$.
We then regularize the dimension of $\mathbf{y}$ from $D$ to $\bar{D}$, and
include an extra factor of $(-t_{0})^{D-\bar{D}}$ in order to ensure that the
overall dimension remains correct.

$\displaystyle\int\frac{d^{D-1}\mathbf{p}}{(2\pi)^{D-1}}\frac{1}{|p|^{2s_{3}+2s_{5}}|p+k|^{2s_{4}+2s_{6}}}=(-t_{0})^{D-\bar{D}}\int
d^{\bar{D}-1}\mathbf{y}\
e^{-i\mathbf{k}\cdot\mathbf{y}}f_{2s_{3}+2s_{5}}(\mathbf{y})f_{2s_{4}+2s_{6}}(\mathbf{y})$
$\displaystyle=$
$\displaystyle\frac{k^{D-1-2(s_{3}-s_{4}-s_{5}-s_{6})}\Gamma\left(s_{3}+s_{4}+s_{5}+s_{6}-\frac{2D-\bar{D}-1}{2}\right)\Gamma\left(\frac{D-1}{2}-s_{3}-s_{5}\right)\Gamma\left(\frac{D-1}{2}-s_{4}-s_{6}\right)}{(4\pi)^{\frac{2D-\bar{D}-1}{2}}(-kt_{0})^{\bar{D}-D}\Gamma\left(s_{3}+s_{5}\right)\Gamma\left(s_{4}+s_{6}\right)\Gamma\left(D-1-(s_{3}+s_{4}+s_{5}+s_{6})\right)}\,.$

The parameter $D-\bar{D}$ now allows us more freedom when choosing the
contour, and we are thus able to avoid the ambiguities discussed above. In
contrast with the case of the quartic loop, in which integration over the loop
momentum yields a delta function, directly enforcing a conservation law, in
the cubic case the loop integration leads to more complicated pole structures.
The difference can be understood similarly to what happens in flat space,
where the massless quartic loop is a vacuum massless bubble and does not
contribute to the self-energy when computed using dimensional regularization.
In our de Sitter case, for the specific example above, we can regulate the
cubic loop appropriately by choosing $D=4+2\kappa$ and $\bar{D}=4$.

We now turn to the time integrations in our expressions (5.35) and (5.35).
Focusing first on the time integration in (5.35), we obtain:

$\int_{-\infty}^{t_{0}}dt_{1}\int_{-\infty}^{t_{1}}dt_{2}(-t_{1})^{\alpha}(-t_{2})^{\beta}=\int_{-\infty}^{t_{0}}dt_{1}\frac{(-t_{1})^{\alpha+\beta+1}}{\beta+1}\
.$ (5.40)

We then take the late time limit $t_{0}\to 0$, so that the integration over
$t_{1}$ becomes a $\delta$ function, giving

$\lim_{t_{0}\to
0}\int_{-\infty}^{t_{0}}dt_{1}\frac{(-t_{1})^{\alpha+\beta+1}}{\beta+1}=\frac{2i\pi\delta\left(\alpha+\beta+2\right)}{\beta+1}\
,$ (5.41)

where $\alpha=-2s_{1}-2s_{3}-2s_{4}+\frac{D-3}{2}$,
$\beta=-2s_{2}-2s_{5}-2s_{6}+\frac{D-3}{2}$, and we require
$\mathfrak{Re}(\beta)<-1$ to ensure that the $t_{2}$ integration converges. If
we were to directly integrate out $s_{1}$ using the $\delta$ function, we
would encounter the same ambiguity that we found when evaluating the quartic
loop case. We therefore once again regulate by introducing a parameter
$\epsilon$,

$\displaystyle(-t_{0})^{-4\epsilon}\int_{-\infty}^{t_{0}}dt_{1}\int_{-\infty}^{t_{1}}dt_{2}(-t_{1})^{\alpha+2\epsilon}(-t_{2})^{\beta+2\epsilon}\xrightarrow{\lim_{t_{0}\to
0}}\frac{2i\pi\delta\left(\alpha+\beta+4\epsilon+2\right)}{\beta+2\epsilon+1}\
.$ (5.42)

This converges if we require the real part of denominator to be negative,
$\mathfrak{Re}(\beta+2\epsilon+1)<0$. Since $\beta=-2s_{2}-2s_{5}-2s_{6}$,
this requires us to choose the contour to the right of the pole in the
denominator.

In a similar way, if we now focusing instead on the time integration in (5.4),
we obtain:

$\displaystyle(-t_{0})^{-4\epsilon}\int_{-\infty}^{t_{0}}dt_{2}\int_{-\infty}^{t_{2}}dt_{1}(-t_{1})^{\alpha+2\epsilon}(-t_{2})^{\beta+2\epsilon}\xrightarrow{\lim_{t_{0}\to
0}}\frac{2i\pi\delta\left(\alpha+\beta+4\epsilon+2\right)}{\alpha+2\epsilon+1}\
,$ (5.43)

and to ensure that the $t_{1}$ integration is convergent as $t_{1}\to-\infty$,
we require that the denominator $\mathfrak{Re}(\alpha+2\epsilon+1)<0$.

Using these results, we then have

$\displaystyle B_{2c}^{(1)++,>}=-\frac{e^{\frac{\pi
i(2D+1)}{2}}f_{k}^{\ast}(t_{0})^{2}k^{D-1}(-kt_{0})^{2\kappa}}{(4\pi)^{3+(D-1)/2+\kappa}}\int[ds_{i}]\Gamma\left(s_{i}\pm\frac{\nu_{i}}{2}\right)e^{\pi
i(s_{1}+s_{2}+s_{5}+s_{6}-s_{3}-s_{4})}\left(\frac{k}{2}\right)^{-2(s_{1}+\cdots+s_{6})}$
$\displaystyle\times$
$\displaystyle\frac{2i\pi\delta(\alpha+\beta+2+4\epsilon)(-t_{0})^{-4\epsilon}}{\beta+2\epsilon+1}\frac{\Gamma\left(s_{3}+s_{4}+s_{5}+s_{6}-\frac{D-1}{2}\right)\Gamma\left(\frac{D-1}{2}-s_{3}-s_{5}\right)\Gamma\left(\frac{D-1}{2}-s_{4}-s_{6}\right)}{\Gamma\left(s_{3}+s_{5}\right)\Gamma\left(s_{4}+s_{6}\right)\Gamma\left(D-1-(s_{3}+s_{4}+s_{5}+s_{6})\right)}\,,$

where $i=1,\cdots,6$. Integrating $s_{1}$ with respect to the $\delta$
function then yields

$\displaystyle B_{2c}^{(1)++,>}=-\frac{e^{\frac{\pi
i(2D+1)}{2}}f_{k}^{\ast}(t_{0})^{2}}{(4\pi)^{3+\frac{D-1}{2}}\pi^{\kappa}}\left(-\frac{kt_{0}}{2}\right)^{2\kappa-4\epsilon}\int[ds_{i}]\Gamma\left(s_{i}\pm\frac{\nu_{i}}{2}\right)\Gamma\left(\frac{3}{2}+\kappa+2\epsilon-\sum_{i=2}^{6}s_{i}\pm\frac{\nu}{2}\right)$
$\displaystyle\times$ $\displaystyle\frac{e^{\pi
i\left(\frac{3}{2}+\kappa+2\epsilon-2s_{3}-2s_{4}\right)}}{2(\beta+2\epsilon+1)}\frac{\Gamma\left(s_{3}+s_{4}+s_{5}+s_{6}-\frac{3+4\kappa}{2}\right)\Gamma\left(\frac{3+2\kappa}{2}-s_{3}-s_{5}\right)\Gamma\left(\frac{3+2\kappa}{2}-s_{4}-s_{6}\right)}{\Gamma\left(s_{3}+s_{5}\right)\Gamma\left(s_{4}+s_{6}\right)\Gamma\left(3+2\kappa-(s_{3}+s_{4}+s_{5}+s_{6})\right)}\
,$

In the same way, we obtain

$\displaystyle B_{2c}^{(1)++,<}=-\frac{e^{\frac{\pi
i(2D+1)}{2}}f_{k}^{\ast}(t_{0})^{2}}{(4\pi)^{3+\frac{D-1}{2}}\pi^{\kappa}}\left(-\frac{kt_{0}}{2}\right)^{2\kappa-4\epsilon}\int[ds_{i}]\Gamma\left(s_{i}\pm\frac{\nu_{i}}{2}\right)\Gamma\left(\frac{3}{2}+\kappa+2\epsilon-\sum_{i=2}^{6}s_{i}\pm\frac{\nu}{2}\right)$
$\displaystyle\times$ $\displaystyle\frac{e^{\pi
i\left(\frac{3}{2}+\kappa+2\epsilon-2s_{5}-2s_{6}\right)}}{-2(\beta+2\epsilon+1)}\frac{\Gamma\left(s_{3}+s_{4}+s_{5}+s_{6}-\frac{3+4\kappa}{2}\right)\Gamma\left(\frac{3+2\kappa}{2}-s_{3}-s_{5}\right)\Gamma\left(\frac{3+2\kappa}{2}-s_{4}-s_{6}\right)}{\Gamma\left(s_{3}+s_{5}\right)\Gamma\left(s_{4}+s_{6}\right)\Gamma\left(3+2\kappa-(s_{3}+s_{4}+s_{5}+s_{6})\right)}\
,$

where $i=2,\ldots,6$. The denominator in $B_{2c}^{(1)++,<}$ arises from
integrating over the $\delta$-function
$\delta\left(\alpha+\beta+4\epsilon+2\right)$, and the convergence of the
$t_{1}$ integral now requires that
$-\mathfrak{Re}\left(\beta+2\epsilon+1\right)=\mathfrak{Re}\left(\alpha+2\epsilon+1\right)<0$.
We therefore choose the contour to run to the left of the pole.

Notice that the contour integration itself does not involve any momentum
dependence. As a result, the momentum structure of the cubic loop correction
has the late time limit $\mathfrak{Re}\left(f_{k}^{\ast}(t_{0})^{2}\right)$,
which is of the same form as the higher derivative term that is proportional
to $c_{4}^{\prime}$. One can also check that in the late time limit, we have
$f_{k}(t_{0})^{2}=f_{k}^{\ast}(t_{0})^{2}=f_{k}(t_{0})f_{k}^{\ast}(t_{0})$. We
can therefore directly sum the divergent terms, since each of
$B_{2c}^{(1)++},B_{2c}^{(1)--}$ and $B_{2c}^{(1)-+}$ have the same momentum
dependence. What remains is to compute the divergences in the contour
integration that is independent of the momentum.

As discussed in [53], the divergences arise when the left poles overlap with
the right poles. To evaluate these divergences, we introduce regulating
parameters $\epsilon_{i}$ that allow us to choose fixed contour lines that are
parallel to the imaginary axis and that separate the left poles from the right
poles, and then analytically continue the result to the region
$\epsilon_{i}\approx 0$. This procedure has been carefully discussed in [54,
55, 53]. We use the above regularization and choose $\kappa=\epsilon$, and
employ the Mathematica package MB.m [55] to yield the following divergence
structure, where we have summed over
$B_{2c}^{(1)++}=B_{2c}^{(1)++,>}+B_{2c}^{(1)++,<}$ and
$B_{2c}^{--}=B_{2c}^{(1)--,>}+B_{2c}^{(1)--,<}$,

$\left(-\frac{1}{360\sqrt{\pi}\epsilon^{2}}+\frac{\alpha_{1}}{\epsilon}\right)\left(-\frac{kt_{0}}{2}\right)^{-2\epsilon}f_{k}(t_{0})\,f_{k}^{\ast}(t_{0})\,,$
(5.47)

with $\alpha_{1}$ a constant. We provide more details of the contour choices
in the Appendix.

####  5.4.1 $B_{2c}^{(1)-+}$

For the $B_{2}^{(1)-+}$ correlator, both time integrations start from
$-\infty$ and end at $t_{0}$, and the loop divergence is same as in Eq.(5.4):

$\displaystyle B_{2c}^{(1)-+}$ $\displaystyle=$ $\displaystyle
f_{k}(t_{0})f_{k}^{\ast}(t_{0})\int_{-\infty}^{t_{0}}dt_{1,2}\Pi_{i=1}^{6}\int[ds_{i}]\Gamma\left(s_{i}\pm\frac{\nu_{i}}{2}\right)e^{\pi
i(s_{1}+s_{3}+s_{4}-s_{2}-s_{5}-s_{6})}\left(\frac{k}{2}\right)^{-2(s_{1}+s_{2})}$
$\displaystyle\times$
$\displaystyle\int\frac{d^{D-1}\mathbf{p}_{1}}{(2\pi)^{D-1}}(-t_{1})^{-2(s_{1}+s_{3}+s_{4})+\frac{D-3}{2}}(-t_{2})^{-2(s_{2}+s_{5}+s_{6})+\frac{D-3}{2}}\left(\frac{p_{1}}{2}\right)^{-2(s_{3}+s_{5})}\left(\frac{|p_{1}+k|}{2}\right)^{-2(s_{4}+s_{6})}\
.$

Performing the time integrations, and taking the late time limit $t_{0}\to 0$,
we introduce the same regulating parameters as in (5.43) to obtain

$(-t_{0})^{-4\epsilon}\int_{-\infty}^{t_{0}}dt_{1}\int_{-\infty}^{t_{0}}dt_{2}(-t_{1})^{\alpha+2\epsilon}(-t_{2})^{\beta+2\epsilon}\xrightarrow{\lim_{t_{0}\to
0}}2i\pi\delta\left(\alpha+2\epsilon+1\right)2i\pi\delta\left(\beta+2\epsilon+1\right)\
,$ (5.49)

where again (as in the $++$ case),
$\alpha=-2s_{1}-2s_{3}-2s_{4}+\frac{D-3}{2}$,
$\beta=-2s_{2}-2s_{5}-2s_{6}+\frac{D-3}{2}$, and we require the real parts of
$\alpha+2\epsilon+1$ and $\beta+2\epsilon+1$ to be negative to converge.

Using the $\delta$-functions to perform the integrals over $s_{1}$ and
$s_{2}$, the $+-$ correlator is then

$\displaystyle
B_{2c}^{(1)-+}=\frac{f_{k}(t_{0})f_{k}^{\ast}(t_{0})}{4(4\pi)^{\frac{D-1}{2}}\pi^{\kappa}}\left(-\frac{kt_{0}}{2}\right)^{2\kappa-4\epsilon}\Pi_{i=3}^{6}\int[ds_{i}]\Gamma\left(s_{i}\pm\frac{\nu_{i}}{2}\right)\Gamma\left(\frac{3+2\kappa}{4}+\epsilon-
s_{3}-s_{4}\pm\frac{\nu}{2}\right)$
$\displaystyle\Gamma\left(\frac{3+2\kappa}{4}+\epsilon-
s_{5}-s_{6}\pm\frac{\nu}{2}\right)\frac{\Gamma\left(s_{3}+s_{4}+s_{5}+s_{6}-\frac{3+4\kappa}{2}\right)\Gamma\left(\frac{3+2\kappa}{2}-s_{3}-s_{5}\right)\Gamma\left(\frac{3+2\kappa}{2}-s_{4}-s_{6}\right)}{\Gamma\left(s_{3}+s_{5}\right)\Gamma\left(s_{4}+s_{6}\right)\Gamma\left(3+2\kappa-(s_{3}+s_{4}+s_{5}+s_{6})\right)}\
.$

Finally, choosing $\kappa=\epsilon$ and applying the technique described
earlier, we can see that the divergent part of this expression takes the form

$\left(\frac{1}{360\sqrt{\pi}\epsilon^{2}}+\frac{\alpha_{2}}{\epsilon}\right)\left(-\frac{kt_{0}}{2}\right)^{-2\epsilon}f_{k}(t_{0})f_{k}^{\ast}(t_{0})\
.$ (5.51)

With the divergent parts of the $B_{2c}^{(1)++}$, $B_{2c}^{(1)--}$, and
$B_{2c}^{(1)-+}$ correlators in hand, we can now sum them and observe that the
$\frac{1}{\epsilon^{2}}\left(-\frac{kt_{0}}{2}\right)^{-2\epsilon}$
divergences cancel, leaving only
$\frac{1}{\epsilon}\left(-\frac{kt_{0}}{2}\right)^{-2\epsilon}$-type
divergences. As in flat spacetime, we can absorb the $1/\epsilon$ divergence
into the coefficient $c_{4}^{\prime}$ and treat the higher derivative term as
a counter term. Thus, the 1-loop correction to the 2-point equal time
correlation function in the late time limit is

$\left\langle\phi(t_{0},\mathbf{k})\phi\left(t_{0},\mathbf{k}^{\prime}\right)\right\rangle_{\text{1-loop}}=f_{\mathbf{k}}(t_{0})f_{\mathbf{k}}^{*}\left(t_{0}\right)\log\left(-\frac{kt_{0}}{2}\right)\,.$
(5.52)

Using this, the full equal time correlator
$\braket{\Phi(t_{0},\mathbf{x}),\Phi(t_{0},\mathbf{x}^{\prime})}$ is therefore
given by,

$\displaystyle\left\langle\Phi(t_{0},\mathbf{k})\Phi\left(t_{0},\mathbf{k}^{\prime}\right)\right\rangle$
$\displaystyle=\frac{C^{2\Delta}}{(-t_{0})^{2\Delta}}\left[1+\frac{1}{C^{D-2}}(-t_{0})^{D-2}\left(\left\langle\phi(t_{0},\mathbf{k})\phi\left(t_{0},\mathbf{k}^{\prime}\right)\right\rangle_{\text{0-loop}}+\left\langle\phi(t_{0},\mathbf{k})\phi\left(t_{0},\mathbf{k}^{\prime}\right)\right\rangle_{\text{1-loop}}+\cdots\right)\right]$
$\displaystyle=\frac{C^{2\Delta}}{(-t_{0})^{2\Delta}}\left[1+\frac{f_{k}(t_{0})f_{k}^{\ast}(t_{0})}{C^{D-2}}(-t_{0})^{D-2}\left(1+\frac{const.}{C^{2}}\log\left(-\frac{kt_{0}}{2}\right)+\cdots\right)\right]\,.$
(5.53)

Now, taking the late-time limit of the mode functions yields

$f_{k}(t_{0})f_{k}^{\ast}(t_{0})=\frac{(-t_{0})\left(-\frac{1}{2}kt_{0}\right)^{-D-1}}{4\pi\sin\left(\frac{D+1}{2}\pi\right)^{2}\Gamma\left(\frac{1-D}{2}\right)^{2}}\sum_{m=0}c_{m}\left(-k^{2}t_{0}^{2}\right)^{m}\
,c_{m}=\frac{\left(-\frac{D}{2}\right)_{m}}{m!\left(\frac{1-D}{2}\right)_{m}\left(-D\right)_{m}}\
,$ (5.54)

where $(a)_{m}=a(a+1)(a+2)\cdots(a+m-1)$ is the Pochhammer symbol.
Furthermore, using the Fourier identity

$\displaystyle\int\frac{d^{d}\mathbf{k}}{(2\pi)^{d}}\frac{e^{i\mathbf{k}\cdot\left(\mathbf{x}-\mathbf{x}^{\prime}\right)}}{\mathbf{k}^{2\alpha}}=B_{d}(2\alpha)\left(\frac{|\mathbf{x}-\mathbf{x}^{\prime}|}{2}\right)^{2\alpha-d},\quad
B_{d}(2\alpha)=\frac{\pi^{d/2}}{(2\pi)^{d}}\frac{\Gamma\left(\frac{d-2\alpha}{2}\right)}{\Gamma(\alpha)}\
,$ (5.55)

we obtain

$\displaystyle\left\langle\phi\left(t_{0},\mathbf{x}\right)\phi\left(t_{0},\mathbf{x}^{\prime}\right)\right\rangle_{0-\mathrm{loop}}=\frac{(-t_{0})|\mathbf{x}-\mathbf{x}^{\prime}|^{2}}{4\pi\sin\left(\frac{D+1}{2}\pi\right)^{2}\Gamma\left(\frac{1-D}{2}\right)^{2}}\left(-\frac{1}{2}t_{0}\right)^{-D-1}\sum_{m=0}c_{m}B_{d}(2\alpha)\left(\frac{4t_{0}^{2}}{|\mathbf{x}-\mathbf{x}^{\prime}|^{2}}\right)^{m}\
,$

with $2\alpha=D+1-2m$ and $d=D-1$. Fom this, one can see that in coordinate
space the correlator is a function of the cross ratio $\xi$ as defined in
(5.1).

Therefore, we finally obtain

$\left\langle\Phi\left(t_{0},\mathbf{x}\right)\Phi\left(t_{0},\mathbf{x}^{\prime}\right)\right\rangle_{0-\text{loop}}=\frac{C^{2\Delta}}{(-t_{0})^{2\Delta}}\left[1+\frac{2^{D+3}}{C^{D-2}}\xi\sum_{m=0}c_{m}B_{d}(2\alpha)\left(\frac{1}{\xi}\right)^{m}\right]\
,$ (5.57)

which is consistent with the form of the correlator for a defect CFT as in
(5.1). The effective field theory approach thus allows us to read off the
arbitrary function $F(\xi)$ to any given order.

For the 1-loop correlator, the logarithmic contribution can be viewed as the
next-to-leading order expansion of
$\left(-\frac{kt_{0}}{2}\right)^{\epsilon}$. Thus, the loop contribution after
performing a Fourier transform would take a similar form as the tree level
expansion, with a $1/C^{2}$ suppression and some changes to the coefficients.

##  6 Conclusions

We have explored some systematics of the effective field theory of conformal
symmetry breaking, which describes fluctuations around symmetry-breaking VEVs.
With this effective theory, correlators can be computed systematically as an
expansion in inverse powers of the symmetry-breaking parameter.

As an example, we began with the computation of the leading terms in the EFT
expansion of the 2 point function in a theory in which conformal symmetry in
broken to Poincaré symmetry. The EFT gives an expansion which is valid at long
distances, complementary to the short distance operator product expansion.

We then studied the breaking that occurs due to a space-like defect in a
Lorentzian CFT; a setup that is relevant for early universe scenarios in which
the reheating surface can be thought of as a conformal defect. We expanded the
EFT around the time-dependent symmetry-breaking VEV and computed the
correlation functions of the fluctuations. Because the background is time-
dependent, we computed the in-in correlators, which involve time integrations
from the infinite past to the space-like defect surface. To compute the next-
to-leading order corrections to the correlators, one must evaluate the loop
diagrams from cubic and quartic interactions. The time integrations in these
loop diagrams are in principle extremely complicated, and so we found it
convenient to make use of Mellin-Barnes representations for the mode
functions. On the CFT side, this symmetry-breaking pattern demands that the
general form of the 2-point correlator takes the form of an arbitrary function
$F$ of the cross-ratio $\xi$. By Fourier transforming, we were able to use our
EFT expansion to provide a series of terms for this arbitrary function .

While we have demonstrated the usefulness of this technique in specific
examples, the idea should be generalizable in a number of ways. For example,
although we limited our calculations to 2-point functions, the EFT can in
principle by used to compute any correlator in a long distance expansion. In
particular, it would be interesting to compute the 3-point correlator in this
our example, which would provide information about non-gaussianity in the
pseudo-conformal universe model. Similarly, while we chose to apply our
methods to the example of a spacelike defect in a Lorentzian CFT, other
maximally symmetric defects in various signatures should work similarly. It
would be interesting to extend our calculations to the case of higher co-
dimension defects, and perhaps even to the case in which there is mixing
between space-like and time-like defects. Another interesting extension would
be to the case with multiple fields $\Phi_{i}$ acquiring VEVs
$\Phi_{i}=C_{i}^{\Delta}/(-t)^{\Delta}$, where spectator fields would live in
different backgrounds and yield different power spectrums. Finally, it may
also be worth exploring the subtleties in the special cases $\Delta=0$ and
$D\leq 2$ that we have bypassed in this work.

##### Acknowledgments:

We would like to thank Austin Joyce for helpful conversations and comments. QL
would like to thank Donggang Wang for helpful suggestions at early stage, and
Sam S.C. Wong for useful discussion. KH acknowledges support from DOE grant
DE-SC0009946 and from Simons Foundation Award Number 658908. The work of QL
and MT is supported in part by US Department of Energy (HEP) Award DE-
SC0013528. MT was also supported in part by the Simons Foundation Origins of
the Universe Initiative, grant number 658904.

##  Appendix A Details of the contour choice

Here we provide more details of how to obtain Eq. (5.47). For a given contour
integral over $N$ variables $s_{i}$,

$\displaystyle\int\frac{ds_{1}}{2\pi i}\cdots\int\frac{ds_{N}}{2\pi
i}\prod_{i}\frac{\Gamma\left(a_{i}+\sum_{k}b_{ik}s_{k}\right)}{\Gamma\left(c_{i}+\sum_{k}d_{ik}s_{k}\right)}\
,$ (A.1)

the signs of $b_{ik}$ determine whether the Gamma functions result in left
poles or right poles. We choose the contours for the above integrals as fixed
straight lines that are parallel to the imaginary axis and that intersect with
the real axis at $\mathcal{C}_{i}$. These well-defined contours should
separate the left poles and the right poles, which is equivalent to requiring

$\displaystyle a_{i}+\sum_{k}b_{ik}\mathcal{C}_{k}>0\ ,\forall i\ .$ (A.2)

However, this may not be the case, and we may be faced with a situation in
which the left poles overlap with the right poles, and we can no longer define
the contour in this simple way. In this case, we can introduce a set of
regulating parameters $\epsilon_{l}$ into the Gamma functions,

$\displaystyle\int\frac{ds_{1}}{2\pi i}\cdots\int\frac{ds_{N}}{2\pi
i}\prod_{i}\frac{\Gamma\left(a_{i}+\sum_{k}b_{ik}s_{k}+\sum_{l}e_{il}\epsilon_{l}\right)}{\Gamma\left(c_{i}+\sum_{k}d_{ik}s_{k}+\sum_{l}f_{il}\epsilon_{l}\right)}\
,$ (A.3)

so that the conditions (A.2) are now satisfied, and then perform an analytic
continuation back to $\epsilon_{l}\rightarrow 0$ afterwards.

This analytic continuation procedure is implemented in the Mathematica package
MB.m available from [55]. Unfortunately, we cannot apply the Mathematica
package directly to our contour integrals, e.g. (5.4), because in our case we
have an extra single pole in the denominator arising from the time
integration. The convergence of the time integration requires that
$s_{2}+s_{5}+s_{6}-\frac{3}{4}-\frac{3\epsilon}{2}>0$ in (5.4), and
$s_{2}+s_{5}+s_{6}-\frac{3}{4}-\frac{3\epsilon}{2}<0$ in (5.4). This changes
the locations of the contour lines $\mathcal{C}_{i}$. Since there is no
momentum dependence in the contour integral, we are free to close the contour
to either the right side or the left side. However, as shown in figure 2 $(a)$
and $(b)$, taking (5.4) as an example, closing the contour to the left does
not encircle the single denominator pole, while closing the contour to the
right does include it. In principle, one can choose to close the contour on
either side, but the convention of the Mathematica package is to always choose
to close the contour to the right. Therefore, the expression for
$B_{2c}^{(1)++,>}$ should not involve the residue at this single pole, while
the expression for $B_{2c}^{(1)++,<}$ should contain it. In other words, the
actual contour choices are those shown in Fig.(2) $(a)$ and $(c)$.

Moreover, since the algorithm is not written to recognize this single pole, it
is necessary to first rewrite it as

$\frac{1}{-2s_{2}-2s_{5}-2s_{6}+\frac{3}{2}+3\epsilon}=\frac{\Gamma\left(-2\left(\frac{3}{4}+\frac{3\epsilon}{2}-s_{2}-s_{5}-s_{6}\right)\right)}{\Gamma\left(-2\left(\frac{3}{4}+\frac{3\epsilon}{2}-s_{2}-s_{5}-s_{6}\right)+1\right)}\
,$ (A.4)

which generates extra fictitious poles at
$-2\left(\frac{3}{4}+\frac{3\epsilon}{2}-s_{2}-s_{5}-s_{6}\right)=-n$ for non-
negative integers $n$ from the numerator in (A.4). One must then remove these
fictitious terms by subtracting the residue at
$1/(-2s_{2}-2s_{5}-2s_{6}+\frac{3}{2}+3\epsilon)$,

$\displaystyle B_{2c}^{(1)++,>\odot}=-\frac{e^{\frac{\pi
i(2D+1)}{2}}f_{k}^{*}\left(t_{0}\right)^{2}}{(4\pi)^{3+\frac{D-1}{2}}\pi^{\kappa}}\left(-\frac{kt_{0}}{2}\right)^{2\kappa-4\epsilon}\int\left[ds_{i}\right]\Gamma\left(s_{i}\pm\frac{\nu_{i}}{2}\right)\Gamma\left(\frac{3+2\kappa}{4}+\epsilon-
s_{3}-s_{4}\pm\frac{\nu}{2}\right)$ (A.5) $\displaystyle\times$
$\displaystyle\Gamma\left(\frac{3+2\kappa}{4}+\epsilon-
s_{5}-s_{6}\pm\frac{\nu}{2}\right)\frac{\Gamma\left(s_{3}+s_{4}+s_{5}+s_{6}-\frac{3+4\kappa}{2}\right)\Gamma\left(\frac{3+2\kappa}{2}-s_{3}-s_{5}\right)\Gamma\left(\frac{3+2\kappa}{2}-s_{4}-s_{6}\right)}{\Gamma\left(s_{3}+s_{5}\right)\Gamma\left(s_{4}+s_{6}\right)\Gamma\left(3+2\kappa-\left(s_{3}+s_{4}+s_{5}+s_{6}\right)\right)}$
$\displaystyle e^{\pi
i\left(\frac{3}{2}+\kappa+2\epsilon-2s_{3}-2s_{4}\right)}\,,$

where $i=3,4,5,6$. However, the $B_{2c}^{(1)++,<}$ contour integral does not
face this issue, since the pole should be included when we close the contour
to the right, as shown in figure 2 (d).

Figure 2: The different contour choices. The green dot represents the simple
pole in the denominator, the red dots represent the right poles arising from
$\Gamma(a_{i}+\sum_{k}b_{ik}s_{k})$ when $b_{ik}$ is negative, and the blue
dots represent the left poles when $b_{ik}$ is positive. All the poles lie on
the real axis. $B_{2c}^{(1)++,>}$ requires the contour line to lie to the
right of the simple pole, as shown in $(a)$ and $(b)$. The $B_{2c}^{(1)++,<}$
integral requires the contour line to lie to the left of the simple pole, as
shown in $(c)$ and $(d)$.

For $B_{2c}^{(1)++,>}$, the contour choices can be made as follows,

$\epsilon=\frac{163}{44},\ \mathcal{C}_{2}=\frac{61}{48},\
\mathcal{C}_{3}=\frac{553}{384},\ \mathcal{C}_{4}=\frac{131}{96},\
\mathcal{C}_{5}=\frac{695}{384},\ \mathcal{C}_{6}=\frac{191}{96}\ .$ (A.6)

The sum of the residues then gives us that the leading order divergence is
$-\frac{1}{1152\sqrt{\pi}\epsilon^{2}}$. Notice that the fictitious pole
Eq.(A.5) gives a divergence $-\frac{1}{2880\sqrt{\pi}\epsilon^{2}}$, which we
subtract so that the divergence from $B_{2c}^{(1)++,>}$ is given by

$-\frac{1}{1152\sqrt{\pi}\epsilon^{2}}-\left(-\frac{1}{2880\sqrt{\pi}\epsilon^{2}}\right)=-\frac{1}{1920}\frac{1}{\sqrt{\pi}\epsilon^{2}}\
.$ (A.7)

The contour choice for $B_{2c}^{(1)++,<}$, which satisfies the condition
$s_{2}+s_{5}+s_{6}-\frac{3}{4}-\frac{3\epsilon}{2}<0$, can be made as

$\epsilon=\frac{327}{128},\ \mathcal{C}_{2}=\frac{643}{512},\
\mathcal{C}_{3}=\frac{253}{128},\ \mathcal{C}_{4}=\frac{741}{512},\
\mathcal{C}_{5}=\frac{163}{128},\ \mathcal{C}_{6}=\frac{987}{512}\ .$ (A.8)

Evaluating the sum of the residues gives us
$-\frac{1}{1152\sqrt{\pi}\epsilon^{2}}$, and summing both $B_{2c}^{++,<}$ and
$B_{2c}^{++,>}$ then yields

$-\frac{1}{1920}\frac{1}{\sqrt{\pi}\epsilon^{2}}-\frac{1}{1152}\frac{1}{\sqrt{\pi}\epsilon^{2}}=-\frac{1}{720}\frac{1}{\sqrt{\pi}\epsilon^{2}}\
.$ (A.9)

Since $B_{2c}^{(1)--}$ is just the complex conjugation of $B_{2c}^{(1)++}$, we
can safely double the above divergence and obtain that the leading order
divergence for $B_{2c}^{(1)++}+B_{2c}^{(1)--}$ is
$-\frac{1}{360\sqrt{\pi}\epsilon^{2}}$.

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