On the IR divergences in de Sitter space:
loops, resummation and the semi-classical wavefunction

Sebastián Céspedes1, Anne-Christine Davis2 and Dong-Gang Wang2

1 Department of Physics, Imperial College, London, SW7 2AZ, UK

2 Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

Abstract

In this paper, we revisit the infrared (IR) divergences in de Sitter (dS)
space using the wavefunction method, and explicitly explore how the
resummation of higher-order loops leads to the stochastic formalism. In light
of recent developments of the cosmological bootstrap, we track the behaviour
of these nontrivial IR effects from perturbation theory to the non-
perturbative regime. Specifically, we first examine the perturbative
computation of wavefunction coefficients, and show that there is a clear
distinction between classical components from tree-level diagrams and quantum
ones from loop processes. Cosmological correlators at loop level receive
contributions from tree-level wavefunction coefficients, which we dub
classical loops. This distinction significantly simplifies the analysis of
loop-level IR divergences, as we find the leading contributions always come
from these classical loops. Then we compare with correlators from the
perturbative stochastic computation, and find the results there are
essentially the ones from classical loops, while quantum loops are only
present as subleading corrections. This demonstrates that the leading IR
effects are contained in the semi-classical wavefunction which is a
resummation of all the tree-level diagrams. With this insight, we go beyond
perturbation theory and present a new derivation of the stochastic formalism
using the saddle-point approximation. We show that the Fokker-Planck equation
follows as a consequence of two effects: the drift from the Schrödinger
equation that describes the bulk time evolution, and the diffusion from the
Polchinski’s equation which corresponds to the exact renormalization group
flow of the coarse-grained theory on the boundary.

###### Contents

  1. 1 Introduction
  2. 2 The Wavefunction Method: Classical vs Quantum
    1. 2.1 Classical and Quantum Wavefunction Coefficients
    2. 2.2 Classical Loops in Cosmological Correlators
  3. 3 IR Divergences in the Perturbative Computation
    1. 3.1 Loops and IR Divergences
    2. 3.2 IR Divergences at Tree Level
    3. 3.3 One-Loop Structure of IR-Divergent Correlators
    4. 3.4 Multi-Loop Diagrams and Leading Logarithms
  4. 4 What does the Stochastic Formalism Resum?
    1. 4.1 Long Modes Correlations from the Wavefunction
    2. 4.2 Correlators in the Perturbative Stochastic Formalism
  5. 5 Beyond Perturbation Theory
    1. 5.1 Diffusion as the RG Flow on the Boundary
    2. 5.2 Fokker-Planck = Schrödinger + Polchinski
  6. 6 Summary and Outlook
  7. A In-In Computation of One-Loop Corrections
  8. B Probability Distribution from the Wavefunction
  9. C Stochastic Formalism with Two Fields
  10. D Corrections to the Fokker-Planck equation

##  1 Introduction

In the past several decades, great efforts have been made in understanding the
infrared (IR) divergences in de Sitter (dS) space. At the conceptual level,
this issue has important implications, such as for the radiative stability of
dS spacetime, the measure problem of eternal inflation, and also the
holographic description for our Universe. Furthermore, the IR effects in dS
are of phenomenological interest for inflationary cosmology. There, light
scalars are supposed to be responsible for the quantum origin of cosmic
structures, and correlation functions with nontrivial IR behaviour at the end
of inflation may become primary targets that can be tested in upcoming
cosmological observations.

In our current understanding, these IR divergences arise in the perturbation
theory of interacting light scalars. For quantum field theories in a fixed dS
background, it is widely recognized that correlators of these fields may
exhibit secular growth on superhorizon scales [1, 2, 3, 4, 5, 6, 7, 8, 9].
These divergences already show up at tree level, and become more and more
singular for higher-order loops, which in due course invalidate the
perturbative analysis. Therefore in order to check whether final results are
IR-safe or not, in principle one needs to go beyond perturbation theory and
perform a resummation of all the higher-order loop diagrams, though in
practice this seems to be out of reach. Meanwhile, in parallel with the field-
theoretic computations, a non-perturbative approach called the stochastic
formalism was proposed by Starobinsky et al., which applies the Fokker-Planck
equation to describe the probability distribution of long wavelength
perturbations during inflation [10, 11]. There it has been shown that
ultimately, an equilibrium state will be reached for long modes on
superhorizon scales. Thus final correlation functions remain IR-finite. Since
then, the stochastic formalism has been studied as a solution of the IR
divergences in dS using various methods, see [12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23] for recent examples.

In spite of the extensive analysis, some basic questions remain about the
stochastic formalism. The first one is about the explicit connection with the
field-theoretic approaches. While the equilibrium state emerges as a solution
of the Fokker-Planck equation in the non-perturbative regime, it is still
unclear how this description is related to the IR divergences in perturbative
computations, or in other words, what does the stochastic formalism actually
resum.111The pertubative stochastic computation was found to match 1-loop
corrections from in-in formalism [24, 25], while more recently Ref. [13]
showed that the leading divergences from higher-loop diagrams agree with the
stochastic results at the corresponding order. We refer the reader to the end
of Section 4 for detailed discussions. Since the computation becomes more and
more complicated as we increase the order of loops, at first sight it looks
hopeless to provide an exact answer to this question. Meanwhile, the
stochastic formalism is essentially a coarse-grained description for long
wavelength modes where short wavelength ones are being smoothed out. This
recalls to the similar idea of the Wilsonian effective field theory:
integrating out high energy modes leads to a low-energy description which
changes with the ultraviolet (UV) cutoff scale as the renormalization (RG)
group flow. It is very tempting to see how the story in cosmology can be
phrased in terms of this Wilsonian picture.

Another interesting question concerns the situation where we have multiple
interacting light scalars in de Sitter space. For studies of cosmic inflation,
this corresponds to the multi-field models, where the inflaton is not the only
light scalar degree of freedom. Unlike the inflaton field, the additional
light scalars have less constraints and typically lead to IR divergences with
various forms [26]. Recently, a bootstrap analysis was performed within
perturbation theory, and it was shown that the conversion process in multi-
field models is actually described by the IR-singular secular growth in the
field-theoretic approach [27]. This in the end provides a rigorous derivation
for the well-known prediction of local non-Gaussianity from multi-field
inflation. Meanwhile a non-perturbative treatment using the stochastic
formalism is presented in Ref. [28] (See also [29]). It remains to be seen how
new phenomenologies of primordial non-Gaussianity would appear in the non-
perturbative regime of the multi-field system.

Motivated by these questions, the purpose of this work is less of making
conceptually new discoveries on this extensively-discussed topic, rather than
trying to gain a better understanding using the recently developed
wavefunction approach for cosmological perturbations. Our investigation is
initiated by the rapid development of the cosmological bootstrap in the past
several years, where a timeless description of cosmological observables is
being constructed by imposing consistency requirements [30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 27]. A central object in the bootstrap analysis is the
wavefunction of the Universe $\Psi[\phi({\bf x}),t]$, which is a functional of
boundary field fluctuations [57, 58, 59]. For a quantum system, the
probability distribution is famously related to the wavefunction as

$P=|\Psi|^{2}\leavevmode\nobreak\ .$ (1.1)

Meanwhile the Fokker-Planck equation is a description for the evolution of the
probability distribution of long wavelength perturbations. In this sense, for
the purpose of building the link between the field-theoretic and stochastic
approaches, the wavefunction of the Universe provides the most obvious object
to look at. This line of thinking has been explored in Ref. [12], where a
systematic derivation of the stochastic formalism is presented using the
wavefunction method.

In this paper, we adopt a similar approach to track the behaviour of IR-
divergent correlators from the perturbation theory to the non-perturbative
regime. Our analysis highlights the role of the semi-classical wavefunction,
with which we simplify the issue of IR divergences and perform a more detailed
analysis through concrete examples and explicit computations. In perturbation
theory, we will propose the notion of classical loops, and show that for
diagrams beyond tree level, these are the leading contributions to loop-level
correlators with IR divergences. Then through explicit comparison with the
perturbative stochatic computation, we will further identify the stochastic
formalism as actually being a resummation of these classical loop
contributions, while the quantum loops only lead to subleading corrections.

Beyond perturbation theory, we further explore the implications of the semi-
classical wavefunction, which now by itself is a resummation of all the tree-
level contributions. Considering a coarse-grained theory on the boundary, we
find that its exact RG flow is described by the Polchinski equation with an
extra boundary term. This non-perturbative RG equation leads to the diffusion
term in the Fokker-Planck equation, while the drift term results from the
Schrödinger equation which describes the time evolution of the semi-classical
wavefunction.

The structure of the paper is organized as follows. In Section 2, we first
briefly review the wavefunction method for computing cosmological correlation
functions in perturbation theory. Here, we specifically make a distinction
between classical and quantum effects, and propose the notion of quantum and
classical loops for the loop-level correlators. After this preparation, we
move to look into IR divergences in de Sitter space. In Section 3, we perform
the perturbative computations from various types of interacting theories of
light scalars. Through the explicit one-loop analysis and generalizations to
multi-loop diagrams, we establish that the leading IR-divergent contributions
to correlators always originate from classical loops. In Section 4, we
explicitly compare the perturbative results from the wavefunction computation
and the stochastic formalism, and identify that the latter approach is
resumming classical loops. In Section 5, we go beyond perturbation theory and
re-derive the Fokker-Planck equation by identifying the diffusion as a
consequence of the boundary exact renormalization group (RG) equation for the
semi-classical wavefunction. We conclude in Section 6 with a summary and
outlook.

More technical details are presented in the Appendices. In Appendix A, we
perform the in-in computation for one-loop IR-divergent correlators, which
match the results from the wavefunction formalism in Section 3.3. In Appendix
B, we use the coarse-grained wavefunction to reproduce the probability
distributions of long wavelength fluctuations within perturbation theory, and
show that the contributions from tree-level wavefunction coefficients satisfy
the Fokker-Planck equation, which provides another justification for the
relation between the stochastic formalism and the semi-classical wavefunction.
In Appendix C, we extend the derivation of the Fokker-Planck equation to a
two-field system. Throughout the paper, we mainly take the convention of
natural units $c=\hbar=1$, with the metric signature $(-,+,+,+)$, except for
special cases (as in Section 2) where we keep $\hbar$ explicit for the
distinction of quantum and classical effects. The bulk fields are denoted by
$\Phi$, $\Sigma$, while $\phi$ and $\chi$ are boundary fluctuations.

##  2 The Wavefunction Method: Classical vs Quantum

Before delving into the resolution of the infrared divergence problem in de
Sitter space, in this section we first outline the wavefunction of the
Universe formalism. Our primary goal here is to derive the Feynman rules
required for perturbative calculations by employing functional quantization in
a rigorous manner. We explicitly delineate the clear demarcation between
classical and quantum processes within this framework, with both types of
processes making contributions to the cosmological correlation functions at
the level of loop diagrams.

The wavefunction method for computing correlation functions offers significant
advantages, both in terms of conceptual clarity and computational efficiency.
For a quantum system, the wavefunction contains all the information about
correlations and is naturally a more primitive object to study. In cosmology,
this has been appreciated in the recent advances of the Cosmological Bootstrap
program, where we have a collection of quantum fields on a fixed de Sitter
background and the objects of interest are correlations on the late-time
boundary of the spacetime. In perturbation theory, it has been shown that the
analytical structures of the wavefunction are nicely manifested as
consequences of unitarity, locality and spacetime symmetry. Meanwhile, at the
practical level, the wavefunction in many cases becomes a much simpler object
for computations. In this section, we review the wavefunction method for
computing correlation functions, and in particular we turn our attention to
examining the loop-level diagrams of the cosmological correlators,
highlighting the distinctions between classical and quantum contributions
there.

In general, for the quantum-mechanical state of a given system
$|{\Psi}\rangle$, the wavefunction is defined in the Heisenberg picture as its
inner product with the basis of field eigenstates $|\phi(t_{0},{\bf
x})\rangle$ at a fixed time $t_{0}$

$\Psi[\phi({\bf x})]\equiv\langle\phi(t_{0},{\bf
x})|{\Psi}\rangle\leavevmode\nobreak\ .$ (2.1)

Typically we are interested in the interacting vacuum state
$|{\Psi}\rangle=|\Omega\rangle$ defined at the infinite past
($t_{i}=-\infty$), and thus the wavefunctional $\Psi[\phi(x)]$ describes the
overlap between the vacuum state and a given field profile $\phi(x)$ at
$t_{0}$. Formally, the wavefunction of this scalar field $\phi$ can be
presented as the following path integral

$\Psi[\phi({\bf x})]=\int\limits_{\begin{subarray}{c}\Phi(t_{0})\,=\,\phi\\\
\Phi(-\infty)\,=\,0\end{subarray}}\mathcal{D}\Phi\exp\left({\frac{i}{\hbar}S[\Phi]}\right)\leavevmode\nobreak\
,$ (2.2)

where we assume the bulk field $\Phi$ goes to zero (vacuum) in the infinite
past, and is given by the profile $\phi({\bf x})$ at the late time $t_{0}$.
Then the path integral sums over all the possible field configurations with
these two fixed boundary conditions. Recall that in QFT textbooks, the path
integrals normally have a final state at the infinite future
($t_{f}\rightarrow+\infty$), the major difference in the wavefunction
formalism is the presence of a fixed future boundary. This description applies
to general FLRW spacetimes, including Minkowski and de Sitter spaces. For the
rest of this section, we shall keep the analysis formal and general, which may
simply extend to other spacetime background and field components as well.

Another thing to notice is that in (2.2) we keep $\hbar$ explicit, for the
convenience of tracking quantum and semi-classical contributions in the
wavefunction method. As is well-known, in the path integral quantization the
distinction between classical and quantum physics can be clearly demonstrated
by taking the $\hbar\rightarrow 0$ limit. There the path integral is dominated
by the configuration $\phi(t,{\bf x})$ which has an extremum with $\delta
S=0$. This configuration precisely leads to the Euler-Lagrange equation
satisfied by a classical field. In standard quantum field theory, one common
lore is that the processes that can be described by the classical equations of
motion give rise to tree-level diagrams, while loop diagrams, which are truly
quantum effects, cannot be generated in classical field theories and are only
caused by higher-order (in $\hbar$) corrections to the classical computation.
This difference is formally demonstrated by the Schwinger-Dyson equation where
the contact terms with $\hbar$ are responsible for generating loop processes.
In the rest of this section, we will clarify that for cosmology, this
distinction between classical and quantum effects is similarly present in the
wavefunction method, and the basic object of interest here is the wavefunction
coefficients.

###  2.1 Classical and Quantum Wavefunction Coefficients

In perturbation theory, one convenient way to study the wavefunction is to
write it as an expansion in powers of the field fluctuations on the boundary.
After the spatial Fourier space transformation, it can usually be expressed as

$\displaystyle\Psi[\phi]=$ $\displaystyle\exp\left[\frac{1}{2}\int_{{\bf
k}_{1},{\bf k}_{2}}\psi_{2}({\bf k}_{1},{\bf k}_{2})\leavevmode\nobreak\
\phi_{{\bf k}_{1}}\phi_{{\bf k}_{2}}+\sum_{n=3}^{\infty}\frac{1}{n!}\int_{{\bf
k}_{1},...,{\bf k}_{n}}\,\psi_{n}({\bf k}_{1},...,{\bf
k}_{n})\leavevmode\nobreak\ \phi_{{\bf k}_{1}}\cdot\cdot\cdot\phi_{{\bf
k}_{n}}\,\right]\,,$ (2.3)

where the momentum integrals are given by

$\int_{\bf k}\equiv\int\frac{{\rm d}^{3}k}{(2\pi)^{3}}\leavevmode\nobreak\ ,$
(2.4)

and $\psi_{2}$, $\psi_{3}$, … are the wavefunction coefficients in Fourier
space. These are the central objects in our analysis, which encode the
statistics of $\phi$ and can also be seen as correlations functions of
boundary CFT operators. In the following we shall derive the Feynman rules for
computing $\psi_{n}$, and focus on the differences in the computations of
tree-level and loop-level processes. For demonstration, we mainly use the
following simple scalar field theory in a general FLRW universe as a concrete
example

$S[\Phi]=\int{\rm d}\eta{\rm d}^{3}x\leavevmode\nobreak\
a(\eta)^{4}\left[-\frac{1}{2}\left(\partial_{\mu}\Phi\right)^{2}-\frac{1}{2}m^{2}\Phi^{2}-\frac{g}{3!}\Phi^{3}\right]\leavevmode\nobreak\
,$ (2.5)

where the spacetime metric is the one with conformal time $\eta$: ${\rm
d}s^{2}=a(\eta)^{2}\left[-{\rm d}\eta^{2}+{\rm d}{\bf x}^{2}\right]$, and we
choose the mass and coupling to be independent of $\hbar$.

#### Tree diagrams

The derivation of Feynman rules for the tree diagrams has been well presented
in the literature. Here we briefly summarize the basic steps, and the readers
may refer to [57, 59, 44, 60] for more details. The starting point is the
saddle-point approximation, where the action is on-shell

$\Psi[\phi]\simeq\exp\left({\frac{i}{\hbar}S[\Phi_{\rm
cl}]}\right)\leavevmode\nobreak\ ,$ (2.6)

with $\Phi_{\rm cl}$ being the solution of the classical equation of motion.
For the example in (2.5), this is given by

$(\square-m^{2})\Phi_{\rm cl}=\frac{g}{2}\Phi_{\rm cl}^{2}\leavevmode\nobreak\
,$ (2.7)

where $\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ is the d’Alembert operator
in the curved spacetime. We refer to the saddle-point approximation in (2.6)
as the semi-classical wavefunction, which will be of central interest in our
analysis. Next we work in the Fourier space aiming to derive the explicit
expressions of wavefunction coefficients in (2.3). As we impose the boundary
condition $\Phi_{\rm cl}(\eta_{0},{\bf k})=\phi_{\bf k}$, formally the on-
shell equation (2.7) can be solved as

$\Phi_{\rm cl}(\eta,{\bf k})=\phi_{\bf
k}K(k,\eta)+\frac{i}{\hbar}\left.\int{\rm
d}\eta^{\prime}G(k;\eta,\eta^{\prime})\frac{\delta S_{\rm
int}}{\delta\Phi_{\bf k}(\eta^{\prime})}\right|_{\Phi=\Phi_{\rm
cl}}\leavevmode\nobreak\ .$ (2.8)

Here the first term is the part that solves the free equation of motion, and
$K(k,\eta)$ is also known as the bulk-to-boundary propagator given by

$\left(\square_{k}-m^{2}\right)\leavevmode\nobreak\
K(k,\eta)=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
with}\leavevmode\nobreak\ \leavevmode\nobreak\
\lim_{\eta\rightarrow\eta_{0}}K(k,\eta)=1\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\
\lim_{\eta\rightarrow-\infty}K(k,\eta)=0\leavevmode\nobreak\ ,$ (2.9)

where $\square_{k}$ is the spatial Fourier transformation of the d’Alembert
operator. The second term treats the interaction as a source, and introduces
the Green’s function $G(k;\eta,\eta^{\prime})$ which satisfies

$\left(\square_{k}-m^{2}\right)\leavevmode\nobreak\
G(k;\eta,\eta^{\prime})=\frac{i\hbar}{a^{4}}\delta(\eta-\eta^{\prime})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm with}\leavevmode\nobreak\
\leavevmode\nobreak\
\lim_{\eta,\eta^{\prime}\rightarrow-\infty}G(k;\eta,\eta^{\prime})=\lim_{\eta,\eta^{\prime}\rightarrow\eta_{0}}G(k;\eta,\eta^{\prime})=0\leavevmode\nobreak\
.$ (2.10)

In classical theories, the appearance of $\hbar$ is not necessary as in the
end (2.8) is $\hbar$-independent. Here it has been introduced in such a way to
match the one in a quantum theory as we shall show very soon. For those
familiar with bulk-to-bulk propagators in the in-in formalism, one nontrivial
difference to notice is that the wavefunction $G$ propagator has a vanishing
boundary condition as $\eta\rightarrow\eta_{0}$.

Then the tree-level wavefunction coefficients can be derived by substituting
the formal solution (2.8) into the saddle-point wavefunction expression (2.6)
iteratively. Explicitly, up to $g^{2}$ order for the example in (2.5), the
action can be written into the following form

$\displaystyle S[\Phi_{\rm cl}]=$ $\displaystyle\int_{{\bf k}_{1},{\bf
k}_{2}}\left[\left.\frac{1}{2}a(\eta)^{2}\Phi_{{\bf
k}_{1}}\partial_{\eta}\Phi_{{\bf
k}_{2}}\right|_{\eta=\eta_{0}}+\frac{1}{2}\int{\rm d}\eta
a(\eta)^{4}\Phi_{{\bf k}_{1}}(\square_{k}-m^{2})\Phi_{{\bf
k}_{2}}\right]{(2\pi)^{3}}\delta^{(3)}({\bf k}_{1}+{\bf k}_{2})$
$\displaystyle-\frac{g}{3!}\int_{{\bf k}_{1},{\bf k}_{2},{\bf k}_{3}}\int{\rm
d}\eta a(\eta)^{4}K(k_{1},\eta)K(k_{2},\eta)K(k_{3},\eta)\phi_{{\bf
k}_{1}}\phi_{{\bf k}_{2}}\phi_{{\bf k}_{3}}{(2\pi)^{3}}\delta^{(3)}({\bf
k}_{1}+{\bf k}_{2}+{\bf k}_{3})$
$\displaystyle+\frac{ig^{2}}{4!\hbar}\int_{{\bf k}_{1},...,{\bf
k}_{4}}\int{\rm d}\eta\int{\rm
d}\eta^{\prime}a(\eta)^{4}a(\eta^{\prime})^{4}\left[K(k_{1},\eta)K(k_{2},\eta)G(s;\eta,\eta^{\prime})K(k_{3},\eta^{\prime})K(k_{4},\eta^{\prime})\right.$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left.{(2\pi)^{3}}\delta^{(3)}({\bf k}_{1}+{\bf k}_{2}+{\bf
s}){(2\pi)^{3}}\delta^{(3)}({\bf k}_{3}+{\bf k}_{4}-{\bf s})+\text{$t$- and
$u$-channels}\right]\phi_{{\bf k}_{1}}\phi_{{\bf k}_{2}}\phi_{{\bf
k}_{3}}\phi_{{\bf k}_{4}}$ $\displaystyle+...\leavevmode\nobreak\ $ (2.11)

Putting it back into (2.6) and comparing with the notation in (2.3), we
identify the tree-level wavefunction coefficients. The first line in (2.11)
comes from the free theory: the boundary term yields the two-point
wavefunction coefficient $\psi_{2}^{\rm
free}=i\hbar^{-1}a(\eta_{0})^{2}\partial_{\eta_{0}}K(k_{1},\eta_{0}){(2\pi)^{3}}\delta^{(3)}({\bf
k}_{1}+{\bf k}_{2})$, while the second term is proportional to the classical
equation of motion and thus goes to zero once the on-shell condition is
imposed. The second line gives the 3-point function $\psi_{3}$ from the
contact diagram shown by the first diagram in Figure 1; the third line
generates the single-exchange 4-point function $\psi_{4}$ as shown by the
second diagram in Figure 1. Continuing this process, we can generate the
wavefunction results for all the tree-level diagrams, such as the 5-point
function from the double-exchange diagram and so on.

Figure 1: (Semi-)classical wavefunction coefficients from tree-level diagrams.

From this explicit derivation, it is clear that all the tree diagrams follow
from the semi-classical wavefunction (2.6) and the formal solution of the
classical equation (2.8). Diagrammatically, we can compute these wavefunction
coefficients in perturbation theory using the following Feynman rules: i)
assign $ig/\hbar$ to each vertex; ii) assign a $K$ propagator to each external
line and a Green’s function $G$ to each internal line; iii) integrate over the
bulk time for all the insertions ${\rm d}\eta\leavevmode\nobreak\
a(\eta)^{4}$. As there is an $\hbar^{-1}$ with each insertion and a $\hbar$
for each internal line, all the tree-level $\psi_{n}$s have a $\hbar^{-1}$,
which is expected from our starting point – the saddle-point approximation
(2.6).

#### Loop diagrams

Notably, the above derivation cannot generate loops. For flat spacetime
scattering amplitudes, it is well-known that tree-level processes can also
arise in classical field theory, while loop diagrams are only present in a
quantum theory. Similar argument is expected in cosmology as well, but has not
been explicitly presented in literature yet. To identify the truly quantum
effects, we need to go beyond the saddle-point approximation (2.6). Now we
explicitly look into the origin of loops in the wavefunction method.

Let’s first consider the functional quantization of a free theory in the
wavefunction formalism. Without imposing the on-shell condition, the path
integral (2.2) can be expressed in the following form

$\Psi_{0}[\phi]=\int\limits_{\begin{subarray}{c}\Phi(t_{0})\,=\,\phi\\\
\Phi(-\infty)\,=\,0\end{subarray}}\mathcal{D}\Phi\exp\left[{\frac{i}{2\hbar}\int_{{\bf
k},{\bf k}^{\prime}}\left(a^{2}\left.\phi_{{\bf k}}\partial_{\eta}\Phi_{{\bf
k^{\prime}}}\right|_{\eta=\eta_{0}}+\int{\rm d}\eta a(\eta)^{4}\Phi_{{\bf
k}^{\prime}}(\square_{k}-m^{2})\Phi_{{\bf
k}}\right)}(2\pi)^{3}\delta^{(3)}({\bf k}+{\bf
k}^{\prime})\right]\leavevmode\nobreak\ ,$ (2.12)

The main difference with the first line in (2.11) is that now the second term
is no longer zero as the on-shell condition becomes invalid in a quantum
theory. While the boundary term is fixed as in classical theories, the
integration over $\Phi$ can be solved as

$\Psi_{0}[\phi]=\exp\left(\frac{1}{2}\int_{{\bf k},{\bf
k}^{\prime}}\psi_{2}^{\rm free}\phi_{\bf k}\delta\phi_{{\bf
k}^{\prime}}\right)\left[{\rm
det}\left(-i(\square_{k}-m^{2})/2\pi\hbar\right)\right]^{-1/2}\leavevmode\nobreak\
.$ (2.13)

Furthermore, using the Gaussian integral, we can compute the two-point
function of the bulk field $\Phi$

$\langle\Phi_{\bf k}(\eta)\Phi_{\bf
k^{\prime}}(\eta^{\prime})\rangle\equiv\frac{\displaystyle\int\mathcal{D}\Phi\Phi_{\bf
k}(\eta)\Phi_{\bf
k^{\prime}}(\eta^{\prime})\exp\left({\frac{i}{\hbar}S_{0}[\Phi]}\right)}{\displaystyle\int\mathcal{D}\Phi\exp\left({\frac{i}{\hbar}S_{0}[\Phi]}\right)}=G(k,\eta,\eta^{\prime})(2\pi)^{3}\delta^{(3)}({\bf
k}+{\bf k}^{\prime})\leavevmode\nobreak\ ,$ (2.14)

where $G(k,\eta,\eta^{\prime})$ is the same with the Green’s function (2.10)
defined in classical theories. As is clear from this new definition, $G$ is
also called the bulk-to-bulk propagator which takes a bulk field from a time
$\eta^{\prime}$ to another time $\eta$. Now it also becomes obvious for our
choice of putting $\hbar$ in (2.10). In the quantum theory, the $G$
propagators carry an $\hbar$ by definition. As an aside, because of the
boundary condition at $\eta_{0}$ in (2.10), this bulk-to-bulk propagator
differs from the one in the in-in formalism, which will become important for
our discussion on the IR divergences later. The bulk-to-boundary propagator
$K$ can be introduced in a similar manner by using the conjugate momentum
$\Pi=a^{2}\partial_{\eta}\Phi$ on the boundary

$\langle\Pi_{\bf k}(\eta_{0})\Phi_{\bf
k^{\prime}}(\eta)\rangle=a(\eta^{\prime})^{2}\partial_{\eta^{\prime}}\langle\Phi_{\bf
k}(\eta^{\prime})\Phi_{\bf
k^{\prime}}(\eta)\rangle|_{\eta^{\prime}=\eta_{0}}=i\hbar
K(k,\eta)(2\pi)^{3}\delta^{(3)}({\bf k}+{\bf k}^{\prime})\leavevmode\nobreak\
,$ (2.15)

where in the second equality we have used the identity
$K(k,\eta)=a(\eta^{\prime})^{2}\partial_{\eta^{\prime}}G(k,\eta^{\prime},\eta)|_{\eta^{\prime}=\eta_{0}}$.

Next, we move to compute wavefunction coefficients in perturbation theory. In
the path integral formalism, they are expressed as the following matrix
element

$\psi_{n}({\bf k}_{1},...,{\bf
k}_{n})\equiv\left.\frac{\delta^{n}\Psi[\phi]}{\delta\phi_{{\bf
k}_{1}}...\delta\phi_{{\bf
k}_{n}}}\right|_{\phi=0}=\frac{(i/\hbar)^{n}}{\Psi[0]}\int\mathcal{D}\Phi\Pi_{{\bf
k}_{1}}(\eta_{0})...\Pi_{{\bf
k}_{n}}(\eta_{0})\exp\left(\frac{i}{\hbar}S[\Phi]\right)\leavevmode\nobreak\
.$ (2.16)

With the presence of interactions, we can perturbatively expand the
exponential function in (2.16) into
$\exp\left(iS_{0}/\hbar\right)\left(1+iS_{\rm int}/\hbar+...\right)$ and
perform the functional integral to produce Wick contractions. This procedure
leads to the Feynman rules for computing wavefunction coefficients. The
contractions between $\Pi$ and $\Phi$ are given by (2.15), which simply means
that the external legs are associated with the bulk-to-boundary propagators.
Then we also find time-ordered correlation functions of the bulk field $\Phi$,
whose contractions yield $G$ propagators in (2.14). At the tree-level, this
derivation is in precise agreement with the Feynman rules in classical
theories, as it should be. However, one expects nontrivial consequences in a
quantum theory, as it also becomes possible to contract bulk fields in such a
way that closed loops can be formed using the bulk-to-bulk propagators.

Figure 2: Quantum wavefunction coefficients from loop-level diagrams.

As a concrete example, let’s take a look at the leading correction to the two-
point wavefunction coefficient in the quantum version of the $\Phi^{3}$ theory
in (2.5). At the $g^{2}$ order, this correction corresponds to a one-loop
process, which is explicitly given by

$\displaystyle\psi_{2}^{\rm 1-loop}=$
$\displaystyle\frac{g^{2}}{(3!)^{2}\hbar^{4}\Psi[0]}\int\mathcal{D}\Phi\Pi_{{\bf
k}}(\eta_{0})\Pi_{{\bf k}^{\prime}}(\eta_{0})\int_{\rm x,y}{\rm d}\eta_{1}{\rm
d}\eta_{2}a(\eta_{1})^{4}a(\eta_{2})^{4}\Phi(\eta_{1},{\bf
x})^{3}\Phi(\eta_{2},{\bf y})^{3}\exp\left(\frac{i}{\hbar}S_{0}[\Phi]\right)$
$\displaystyle=$ $\displaystyle\frac{g^{2}}{(3!)^{2}\hbar^{4}}\int{\rm
d}\eta_{1}{\rm d}\eta_{2}a(\eta_{1})^{4}a(\eta_{2})^{4}\int_{{\bf p}_{1},{\bf
p}_{2},{\bf p}_{3},{\bf q}_{1},{\bf q}_{2},{\bf
q}_{3}}{(2\pi)^{3}}\delta^{(3)}({\bf p}_{1}+{\bf p}_{2}+{\bf
p}_{3}){(2\pi)^{3}}\delta^{(3)}({\bf q}_{1}+{\bf q}_{2}+{\bf q}_{3})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{\displaystyle\times\langle\Pi_{{\bf k}}(\eta_{0})\Pi_{{\bf
k}^{\prime}}(\eta_{0})\Phi(\eta_{1},{\bf p}_{1})\Phi(\eta_{1},{\bf
p}_{2})\Phi(\eta_{1},{\bf p}_{3})\Phi(\eta_{2},{\bf q}_{1})\Phi(\eta_{2},{\bf
q}_{1})\Phi(\eta_{2},{\bf q}_{3})\rangle}\leavevmode\nobreak\ ,$ (2.17)

where in the second line we have performed the Fourier transformation for $\bf
x$ and $\bf y$. Next we can apply the Wick contractions using (2.14) and
(2.15). In addition to the contraction between the bulk field and the boundary
conjugate momentum, there are also two contractions of two $\Phi$ fields which
form a closed loop. In the end the wavefunction coefficient becomes

$\psi_{2}^{\prime\rm 1-loop}=-\frac{g^{2}}{2}\int{\rm d}\eta_{1}{\rm
d}\eta_{2}a(\eta_{1})^{4}a(\eta_{2})^{4}K(k,\eta_{1})K(k,\eta_{2})\int_{\bf
p}G(p,\eta_{1},\eta_{2})G(|{\bf k}+{\bf
p}|,\eta_{1},\eta_{2})\leavevmode\nobreak\ ,$ (2.18)

where the prime means the momentum conserving $(2\pi)^{3}\delta^{(3)}({\bf
k}+{\bf k}^{\prime})$ has been stripped. Keeping doing this computation to
higher orders in perturbation theory, we find higher-loop corrections to
$\psi_{2}$, whose full expression is given by

$\psi_{2}({\bf k},{\bf
k}^{\prime})\equiv\left.\frac{\delta^{2}\Psi[\phi]}{\delta\phi_{{\bf
k}}\delta\phi_{{\bf k}^{\prime}}}\right|_{\phi=0}=\psi_{2}^{\rm
free}+\psi_{2}^{\rm 1-loop}+\psi_{2}^{\rm 2-loop}+...$ (2.19)

As we can see, these corrections to the two-point function cannot arise in the
classical computation using saddle-point approximations and on-shell
conditions. This is because in a classical theory one cannot contract two bulk
fields within a correlator as we have done in (2.1), while for quantum
theories this is allowed through contact terms in the Schwinger-Dyson equation
[61]. Thus the loop-level wavefunction coefficients have a truly quantum
origin.

Now we need to add one more Feynman rule for the computation of wavefunction
coefficients in quantum theories: use multiple bulk-to-bulk propagators to
form loops and integrate over the loop momentum. Another interesting way to
see the quantum origin of loop diagrams is to notice that the loop expansion
in (2.19) is also an expansion in $\hbar$. Since there is an $\hbar$ for each
bulk-to-bulk propagator, for wavefunction coefficients the order of $\hbar$
increases with the number of loops $L$ as $\hbar^{L-1}$, while all the tree-
level wavefunction coefficients are order $\hbar^{-1}$.

In summary, by deriving the Feynman rules from scratch for the wavefunction
method, we identify that classical and quantum effects are associated with
tree-level and loop-level wavefunction coefficients respectively.222Strictly
speaking, the tree-level wavefunction coefficients are due to semi-classical
effects, as the generation of field fluctuations in cosmology is quantum.
Throughout this paper, we do not distinguish “classical” from “semi-
classical”, but just keep in mind that we are analyzing quantum field theories
in curved spacetime. This is in analogy with our understanding for scattering
amplitudes in Minkowski spacetime. For cosmology, the distinction between
classical and quantum effects is somehow obscured in the in-in formalism
computation for correlation functions. One explanation is that, as shown by
the definition (2.1), the wavefunction is an in-out object like the S-matrix
in Minkowski spacetime but with a fixed-time future boundary (see [56, 62, 63]
for elaboration of this perspective). Therefore it is more natural to discuss
quantum and classical origins in the wavefunction coefficients. While the
equal-time cosmological correlators are intrinsically in-in, and thus the
quantum and classical contributions are in general mixed. Meanwhile, for
cosmological observations we are interested in correlation functions at the
end of inflation, which can be related to the late-time measurement in CMB and
large-scale structure surveys. Next, we shall see for loop diagrams how the
combination of classical and quantum wavefunction coefficients lead us to
these physical observables.

###  2.2 Classical Loops in Cosmological Correlators

In perturbation theory, the way to compute equal-time cosmological correlators
of boundary field fluctuations from the wavefunction of the Universe is to
apply the Born rule

$\langle\phi_{{\bf k}_{1}}\cdots\phi_{{\bf
k}_{n}}\rangle=\frac{\displaystyle\int{\cal D}\phi\,\leavevmode\nobreak\
\phi_{{\bf k}_{1}}\cdots\phi_{{\bf
k}_{n}}\left\lvert\Psi[\phi,\eta_{0}]\right\rvert^{2}}{\displaystyle\int{\cal
D}\phi\leavevmode\nobreak\ \left\lvert\Psi[\phi,\eta_{0}]\right\rvert^{2}}\,.$
(2.20)

With the wavefunction coefficients, we can do the Taylor expansion and then
perform the Gaussian integral over the boundary field $\phi$ and derive the
relations with the boundary correlators. At the tree-level, these equations
are simple algebraic ones. For instance, the two-, three- and four-point
correlators of the $\Phi^{3}$ theory in (2.5) are related to the corresponding
wavefunction coefficients via

$\displaystyle\langle\phi_{\bf k}\phi_{-{\bf k}}\rangle^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{1}{2\hskip 1.0pt{\rm Re}\hskip
1.0pt\psi_{2}^{\prime}(k)}\,,$ (2.21) $\displaystyle\langle\phi_{{\bf
k}_{1}}\phi_{{\bf k}_{2}}\phi_{{\bf k}_{3}}\rangle^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{{\rm Re}\hskip 1.0pt\psi_{3}^{\prime}({\bf k}_{1},{\bf
k}_{2},{\bf k}_{3})}{4{\rm Re}\hskip
1.0pt\psi_{2}^{\prime}(k_{1})\leavevmode\nobreak\ {\rm Re}\hskip
1.0pt\psi_{2}^{\prime}(k_{2})\leavevmode\nobreak\ {\rm Re}\hskip
1.0pt\psi_{2}^{\prime}(k_{3})}\,.$ (2.22) $\displaystyle\langle\phi_{{\bf
k}_{1}}\phi_{{\bf k}_{2}}\phi_{{\bf k}_{3}}\phi_{{\bf k}_{4}}\rangle^{\prime}$
$\displaystyle=$ $\displaystyle\frac{{\rm Re}\hskip
1.0pt\psi_{4}^{\prime}({\bf k}_{1},{\bf k}_{2},{\bf k}_{3},{\bf
k}_{4})}{8\prod_{a=1}^{4}{\rm Re}\hskip
1.0pt\psi_{2}^{\prime}(k_{a})}-\langle\phi_{{\bf k}_{1}}\phi_{{\bf
k}_{2}}\phi_{{\bf k}_{3}}\phi_{{\bf k}_{4}}\rangle^{\prime}_{\rm d}\,,$ (2.23)

where the disconnected part has contributions from a product of two $\psi_{3}$

$\langle\phi_{{\bf k}_{1}}\phi_{{\bf k}_{2}}\phi_{{\bf k}_{3}}\phi_{{\bf
k}_{4}}\rangle^{\prime}_{\rm d}=\frac{1}{{8\prod_{a=1}^{4}{\rm Re}\hskip
1.0pt\psi_{2}^{\prime}(k_{a})}}\left[\frac{{\rm Re}\hskip
1.0pt\psi_{3}^{\prime}({\bf k}_{1},{\bf k}_{2},{\bf s})\hskip 1.0pt{\rm
Re}\hskip 1.0pt\psi_{3}^{\prime}(-{\bf s},{\bf k}_{3},{\bf k}_{4})}{{\rm
Re}\hskip 1.0pt\psi^{\prime}_{2}(s)}+\text{$t$- and $u$-channels}\,\right]\,.$
(2.24)

The primes on correlators and $\psi_{n}$ mean that we stripped the momentum-
conserving $\delta$-functions. Therefore, knowing the classical wavefunction
coefficients, such as (2.11), we are able to compute the tree-level
correlation functions.

For loop diagrams, the relation between correlators and $\psi_{n}$ becomes
more complicated, as both classical and quantum wavefunction coefficients are
expected to contribute to the final correlation functions. For the $\Phi^{3}$
theory, the Born rule leads to the following two-point correlator up to
$g^{2}$ order

$\displaystyle\langle\phi_{{\bf k}_{1}}\phi_{{\bf k}_{2}}\rangle$
$\displaystyle=$ $\displaystyle\int\mathcal{D}\phi\leavevmode\nobreak\
\phi_{{\bf k}_{1}}\phi_{{\bf k}_{2}}\exp\Big{\\{}\frac{1}{2}\int_{{\bf
q}_{1},{\bf q}_{2}}\phi_{{\bf q}_{1}}\phi_{{\bf q}_{2}}\leavevmode\nobreak\
2{\rm Re}\left[\psi_{2}+\psi_{2}^{\rm 1-loop}\right]$ (2.25)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{1}{3!}\int_{{\bf q}_{1},{\bf q}_{2},{\bf q}_{3}}\phi_{{\bf
q}_{1}}\phi_{{\bf q}_{2}}\phi_{{\bf q}_{3}}\leavevmode\nobreak\ 2{\rm
Re}\psi_{3}+\frac{1}{4!}\int_{{\bf q}_{1},...,{\bf q}_{4}}\phi_{{\bf
q}_{1}}\phi_{{\bf q}_{2}}\phi_{{\bf q}_{3}}\phi_{{\bf
q}_{4}}\leavevmode\nobreak\ 2{\rm Re}\psi_{4}\Big{\\}}\leavevmode\nobreak\ ,$

where $\psi_{2}$, $\psi_{3}$ and $\psi_{4}$ are the classical ones that can be
read off from the tree-level computation in (2.11), while $\psi_{2}^{\rm
1-loop}$ is the quantum wavefunction coefficient in (2.1). Expanding the
exponential and performing the Gaussian integral over $\phi$, the one-loop
correction then can be explicitly decomposed into three contributions

$\displaystyle\langle\phi_{{\bf k}_{1}}\phi_{{\bf k}_{2}}\rangle_{\rm 1-loop}$
$\displaystyle=$ $\displaystyle\frac{{\rm Re}\hskip 1.0pt\psi^{\rm
1-loop}_{{\bf k}_{1}{\bf k}_{2}}}{2{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(k_{1})\hskip 1.0pt{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(k_{2})}-\frac{1}{8{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(k_{1})\hskip 1.0pt{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(k_{2})}\int_{\bf p}\frac{{\rm Re}\hskip
1.0pt\psi_{4}({{\bf k}_{1},{\bf k}_{2}},{\bf p},-{\bf p})}{{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(p)}$ (2.26) $\displaystyle+\frac{1}{8{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(k_{1})\hskip 1.0pt{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(k_{2})}\int_{\bf p}\left[\frac{{\rm Re}\hskip
1.0pt\psi^{\prime}_{3}({\bf k}_{1},{\bf p},-{\bf p}-{\bf k}_{1}){\rm Re}\hskip
1.0pt\psi_{3}({\bf k}_{2},-{\bf p},{\bf p}+{\bf k}_{1})}{{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(p){\rm Re}\hskip 1.0pt\psi^{\prime}_{2}(|{\bf p}+{\bf
k}_{1}|)}\right.$ $\displaystyle\left.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +\frac{{\rm Re}\hskip
1.0pt\psi^{\prime}_{3}({\bf k}_{1},{\bf k}_{2},-{\bf k}_{1}-{\bf k}_{2}){\rm
Re}\hskip 1.0pt\psi_{3}({\bf k}_{1}+{\bf k}_{2},{\bf p},-{\bf p})}{{\rm
Re}\hskip 1.0pt\psi^{\prime}_{2}(p){\rm Re}\hskip 1.0pt\psi^{\prime}_{2}(|{\bf
k}_{1}+{\bf k}_{2}|)}\right]\leavevmode\nobreak\ .$

From the expression above we can see that the first term has a truly quantum
origin, which cannot be present in any classical field theory. However, the
in-in correlation functions also contains classical loop contributions from
tree-level wavefunction coefficients. For the case here, the two-point
correlator receives contributions from the product of two contact $\psi_{3}$
and the single-exchange $\psi_{4}$ at the one-loop level. These are the
classical loops in cosmological correlators, which can be computed by solving
the classical equation of motion.333We acknowledge insightful discussions with
Enrico Pajer and Santiago Agui Salcedo to clarify this point.

This analysis applies also to higher-order loops. All in all, in this way we
can always distinguish the classical contributions from the quantum ones. In
practice, however, at higher order it becomes more complicated to derive the
relation between wavefunction coefficients and boundary correlators, as more
diagrams would be involved both at the tree level and loop level. For instance
in the $\Phi^{3}$ theory, to compute $\langle\phi\phi\rangle$ at the two-loop
order would also require $\psi_{4}^{\rm 1-loop}$ and triple-exchange
$\psi_{6}$, in addition to the contact $\psi_{3}$ and single-exchange
$\psi_{4}$. Generally speaking, the Born rule (2.20) tells us that for a given
order in perturbation theory, the correlators receive contributions from all
the wavefunction coefficients and their products at the same order. The
schematic form can be found in (3.4), and also see Figure 5 for a specific
example of higher loop diagrams. In any case, the distinction between
classical and quantum contributions to loop diagrams can still be helpful. In
the following sections, we shall take this approach to examine the IR
divergences caused by interacting massless scalars in de Sitter space. As
these divergences are associated with long wavelength modes which become semi-
classical after horizon-exit, we shall see how the distinction between
classical and quantum loops leads to simplifications and interesting insights
about the IR divergences in dS, and in particular clarifies the explicit
connection with the stochastic formalism.

As a concluding remark of this section, we note that in many circumstances for
cosmology, the classical loop contributions can dominate over the quantum
ones, which means that we only need to use the classical equation of motion to
capture the leading effects in the loop computations. This has been noticed
for a long time in the studies of some late-time processes, such as the
formation of large-scale structure and induced gravitational waves, though it
was not fully clear how the loop computations there were related to the ones
in quantum field theory. Using the wavefunction formalism, here we have
performed a systematical analysis to distinguish the (semi-)classical
contributions from the quantum ones. For the previous (semi-)classical
approximations, it would be interesting to understand better their regime of
validity through this approach and evaluate the effects of quantum
corrections.

##  3 IR Divergences in the Perturbative Computation

Within the wavefunction formalism, in this section we turn our attention to
examine the infrared (IR) divergences of de Sitter boundary correlation
functions in perturbation theory. In particular, after a general review of the
subject in Section 3.1, we will first collect some tree-level computation of
wavefunction coefficients with nontrivial IR behaviour in Section 3.2. After
that, in Section 3.3 we shall examine the one-loop structure of IR-divergent
correlators via the wavefunction approach, and identify the distinct
contributions from classical and quantum loops in three types of theories. In
Section 3.4, we extend the one-loop analysis to the general IR-divergent
correlators in higher-order loop diagrams, and find that contributions from
classical loops are always dominant, which significantly simplifies the
identification of the leading logarithmic behaviour in perturbation theory.

###  3.1 Loops and IR Divergences

In perturbative computations, correlation functions in de Sitter space may
keep growing on superhorizon scales. This typically happens with very light
scalars whose amplitudes freeze after horizon exit and thus can cumulatively
source the growth of correlators. As a result, these correlators become
singular towards the future boundary of dS with $\eta_{0}\rightarrow 0$, which
signals the breakdown of perturbation theory. This is famously known as the IR
divergences in dS and has been extensively discussed in the literature. The
appearance of IR divergences is a property of spacetimes with boundaries [64],
and is very similar in AdS (see for instance [65]). In the case of de Sitter
the fact that the boundary is spacelike has further complications.

Meanwhile, as physical observables, cosmological correlators are supposed to
be IR-safe and stable with respect to higher order radiative corrections. Thus
one may expect these IR divergences become finite once we go beyond the
perturbative regime.444In this paper, we restrict our analysis to quantum
field theories in a fixed dS background, and thus neglect backreactions with
dynamical gravity which might become important for the stability of the dS
spacetime. It has been identified as early in [10] that the theory is finite
because it reaches an equilibrium state. The idea behind this approach is that
modes can be split between the long wavelength classical solution and the
short wavelength quantum noise. Heuristically this corresponds to a stochastic
Langevin equation, which can be translated into a non-perturbative Fokker-
Planck equation. By use of these equations one can compute correlation
functions which are now finite, and in particular, time independent at late
times. From a field-theoretic point of view, this approach is expected to
resum all the divergent contributions, especially the higher order loops. To
understand better the resummation, we first taker a closer look at the
perturbative computations. Here let’s briefly summarize the state of the art
on the topic, especially from the boundary perspective of the bootstrap
approach.

The IR divergences in dS already show up in tree-level processes. As we will
show in more detail in Section 3.2, these IR divergences are the secular terms
with logarithmic growth $\log(-k_{\Sigma}\eta_{0})$ that come from the bulk
time integration for each vertex, with $k_{\Sigma}$ normally being the sum of
energies for one vertex. In a realistic scenario, cosmic inflation must end,
thus the conformal time $\eta_{0}$ at the end of inflation provides a natural
IR cutoff to regularize the logarithmic singularity. In this case,
perturbative computations remain valid as long as the logarithmic function
multiplied by the coupling is smaller than one. Recently, a systematical
investigation on the IR behaviour of both contact and exchange diagrams has
been presented in Ref. [27] by using the cosmological bootstrap (see also [66,
67, 68, 69, 70, 45, 71] for earlier studies). There it has been shown that the
secular growth of late-time correlators is described by the boundary conformal
field theory with anomalies, and the IR-divergent correlators from massless
exchange diagrams lead to local-type non-Gaussianities from multi-field
inflation with full singularity structure. Furthermore, it has been noticed
that in the wavefunction approach, the disconnected contributions to exchange
correlators, such as the one shown in (2.23) and (2.24), typically contain
leading IR divergences, while the connected part may be less singular [27].

Beyond tree-level diagrams, higher order IR divergences are expected from
loops. In addition to secular divergences from time integrals, there is
another type of IR divergences that arise from loop momentum integration. This
has long been noticed in the in-in computation [72, 73, 74, 75, 76]. Typically
we may encounter the following loop integral

$\int_{\bf p}\frac{1}{p^{3}}=\frac{1}{2\pi^{2}}\int_{1/L_{\rm
IR}}^{\Lambda}\frac{dp}{p}=\frac{1}{2\pi^{2}}\log\left(\Lambda L_{\rm
IR}\right)\leavevmode\nobreak\ ,$ (3.1)

where we have introduced two cutoff scales for regularization. The UV cutoff
$\Lambda$ becomes essential when we screen out the short wavelength modes to
study the coarse-grained perturbations at large scales. We will elaborate on
this in Sections 4 and 5. Here let’s take a closer look at the IR cutoff,
which is given by the comoving size of the Universe $L_{\rm IR}$ here. This
scale defines the size of the box for the system that we are interested in.
For instance, at late time $t$, the physical size of the Universe is given by
$a(t)L_{\rm IR}$. One way to interpret it is to consider a minimum Fourier
mode $k_{\rm IR}$ which exits the horizon at some initial time $t_{i}$, such
that $k_{\rm IR}=a(t_{i})H\simeq L_{\rm IR}^{-1}$. This initial time may be
seen as the beginning of inflation. Then all the long wavelength perturbations
we are interested in are generated after $t_{i}$ and have $k>k_{\rm IR}$.

As both momentum integration and nested time integration are involved, in
general in-in computations for loop-level correlators are notoriously
difficult. For cases with IR divergences, one can perform computations from
one-loop corrections (see [73, 77, 24, 5] for previous literature and Appendix
A for several examples), but it becomes much more complicated if we consider
the computation of higher loops.555 Some efforts in this direction have been
made in [13]. Meanwhile, one can also tackle the problem using the
wavefunction formalism. Recently it has been realized that loop computations
become simpler in the wavefunction fortmalism, and one can even derive the
general form of the wavefunction coefficients with IR loops at any order in
perturbation theory [12]. In the following we shall go one step further and
show that, the computation for IR-divergent correlators can also be greatly
simplified by using the wavefunction method, and the leading contributions
always come from classical loops.

#### Setup for computation

Our basic object of interest is the wavefunction coefficients $\psi_{n}$ and
we will apply the Feynman rules derived in Section 2. For massless scalars,
the bulk-to-boundary propagator is given by solving (2.9)

$K(k,\eta)=(1-ik\eta)e^{ik\eta}\leavevmode\nobreak\ .$ (3.2)

Then for a free theory, its two-point wavefunction coefficient can be directly
read off from the boundary term in (2.11)

$\psi_{2}^{\rm free}({\bf k},{\bf k}^{\prime})=\frac{ik^{2}}{\hbar
H^{2}\eta_{0}(1-ik\eta_{0})}(2\pi)^{3}\delta({\bf k}+{\bf
k}^{\prime})\leavevmode\nobreak\ ,$ (3.3)

whose real part gives ${\rm Re}\psi_{2}^{\prime\rm free}=-k^{3}/H^{2}\hbar$
and thus generates the standard power spectrum of massless scalars by using
(2.21). The bulk-to-bulk propagator follows from (2.10)

$\displaystyle G(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\hbar
H^{2}}{2k^{3}}\left[(1+ik\eta)(1-ik\eta^{\prime})e^{ik(\eta^{\prime}-\eta)}\Theta(\eta-\eta^{\prime})+(1-ik\eta)(1+ik\eta^{\prime})e^{ik(\eta-\eta^{\prime})}\Theta(\eta^{\prime}-\eta)\right.$
(3.4) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left.-(1-ik\eta)(1-ik\eta^{\prime})e^{ik(\eta^{\prime}+\eta)}\right]\leavevmode\nobreak\
.$

Here as we approach the boundary with $\eta\rightarrow 0$ this propagator
scales as

$\lim_{\eta\rightarrow
0}G(k,\eta,\eta^{\prime})\rightarrow-\frac{i}{3}H^{2}\eta^{3}\leavevmode\nobreak\
(1-ik\eta^{\prime})e^{ik\eta^{\prime}}\leavevmode\nobreak\ .$ (3.5)

Thus, even for massless scalars the wavefunction $G$ propagator quickly decays
on super-horizon scales, instead of becoming constant. This property becomes
important for the analysis of IR divergences in wavefunction coefficients.

As in Section 2 we have clarified the quantum and classical contributions in
the calculation, now we set $\hbar=1$ to avoid clutter. Particularly the
computation will be performed for two massless scalars $\Phi$, $\Sigma$ in the
bulk, with $\phi$ and $\chi$ being their boundary fluctuations
correspondingly. Our analysis mainly focuses on three types of bulk
interactions:

$\frac{\lambda}{4!}\Phi^{4}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{g}{3!}\Phi^{3}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{\alpha}{2}\dot{\Phi}\Sigma^{2}\leavevmode\nobreak\ .$ (3.6)

Let us give a few more words about the choice of these vertices before moving
to particular computations. The first two are commonly expected for
interacting scalars and have been extensively discussed for the studies of IR
divergences in dS. For inflation the scalar field $\phi$ can be the inflaton
itself, and then normally the two couplings $\lambda$ and $g$ are suppressed
by slow-roll parameters. The third interaction may not be so familiar as it
breaks the boost symmetry. Usually this is seen in many multi-field inflation
models where we have a rolling background of the inflaton field, and the
coupling $\alpha$ is typically associated with the field-space curvature [78].
Moreover, as the inflaton here is protected by a shift symmetry, this coupling
may not be necessarily slow-roll suppressed. We shall see later that the time
derivative in this vertex simplifies the analysis of IR divergences, and thus
this type of interaction provides a working example for us to go from
perturbative computation to the non-perturbative regime.

###  3.2 IR Divergences at Tree Level

The tree-level wavefunction coefficients of interacting massless scalars have
already been computed in the literature. Especially for the ones with IR
divergences, the recent analysis using the cosmological bootstrap provides a
systematic classification for their full analytical forms [27], which can be
derived by solving a set of interesting differential equations from anomalous
conformal Ward identities (see also the corresponding analysis of CFT
correlators in momentum space in [66, 67, 68, 69, 70, 71]). Here we mainly
collect the result of tree-level wavefunction coefficients of interacting
massless scalars, from both contact and exchange diagrams.

  * •

$\Phi^{4}$ interaction: the four-point contact diagram leads to

$\displaystyle\psi^{\prime}_{4}$ $\displaystyle=$ $\displaystyle
i\lambda\int_{-\infty}^{\eta_{0}}d\eta
a(\eta)^{4}K(k_{1},\eta)K(k_{2},\eta)K(k_{3},\eta)K(k_{4},\eta)$ (3.7)
$\displaystyle=$
$\displaystyle-\frac{\lambda}{H^{4}}\left[\frac{i}{3\eta_{0}^{3}}+\frac{i}{2\eta_{0}}\sum_{a=1}^{4}k_{a}^{2}+\frac{1}{3}\left(\sum_{a=1}^{4}k_{a}^{3}\right)\log\left(-k_{T}\eta_{0}\right)\right]\leavevmode\nobreak\
,$

where $k_{T}=k_{1}+k_{2}+k_{3}+k_{4}$. Then the corresponding correlation
function is given by the first term in (2.23), while there is no disconnected
terms from $\phi^{4}$ interaction. As the imaginary parts do not contribute,
here the logarithmic term with secular growth gives rise to the IR divergence
in the correlator.

  * •

$\Phi^{3}$ interaction: we first have a three-point contact diagram (see the
first one in Figure 1)

$\displaystyle\psi^{\prime}_{3}$ $\displaystyle=$ $\displaystyle
ig\int_{-\infty}^{\eta_{0}}d\eta
a(\eta)^{4}K(k_{1},\eta)K(k_{2},\eta)K(k_{3},\eta)$ (3.8) $\displaystyle=$
$\displaystyle-\frac{g}{H^{4}}\left[\frac{i}{3\eta_{0}^{3}}+\frac{i}{2\eta_{0}}\sum_{a=1}^{3}k_{a}^{2}+\frac{1}{3}\left(\sum_{a=1}^{3}k_{a}^{3}\right)\log\left(-k_{t}\eta_{0}\right)\right]\leavevmode\nobreak\
,$

where $k_{t}=k_{1}+k_{2}+k_{3}$. The correlator follows from (2.22). Similarly
the IR divergence is caused by the logarithmic secular term. Let’s also
consider the exchange diagram (see the second one in Figure 1) from this
interaction, which leads to the following four-point wavefunction coefficient

$\displaystyle\psi^{\prime}_{4}$ $\displaystyle=$
$\displaystyle-g^{2}\int_{-\infty}^{\eta_{0}}d\eta
d\eta^{\prime}a(\eta)^{4}K(k_{1},\eta)K(k_{2},\eta)G(s,\eta,\eta^{\prime})K(k_{3},\eta^{\prime})K(k_{4},\eta^{\prime})+{\rm
perms.}$ (3.9) $\displaystyle=$
$\displaystyle-\frac{g^{2}}{18H^{6}}\Big{[}(k_{1}^{3}+k_{2}^{3}+s^{3})\log^{2}\left(-E_{L}\eta_{0}\right)+(k_{3}^{3}+k_{4}^{3}+s^{3})\log^{2}\left(-E_{R}\eta_{0}\right)$
$\displaystyle+\text{$t$- and $u$-channels}+{\rm imaginary\leavevmode\nobreak\
part}\Big{]}\leavevmode\nobreak\ ,$

where ${s}=|{\bf k}_{1}+{\bf k}_{2}|$, ${t}=|{\bf k}_{1}+{\bf k}_{3}|$,
${u}=|{\bf k}_{1}+{\bf k}_{4}|$, and $E_{L}=k_{1}+k_{2}+s$,
$E_{R}=k_{3}+k_{4}+s$. In this case, to derive the corresponding correlator,
in to addition the exchange $\psi_{4}$ above, there is also the disconnected
piece in (2.23) with the product of two $\psi_{3}$ in (3.8). Notice that here
these two parts both contribute at the $\log^{2}$ order, thus one needs to
take into account connected and disconnected terms to capture the leading IR-
divergent behaviour in the correlator.

Figure 3: Tree diagrams from the two-field interaction $\dot{\Phi}\Sigma^{2}$.
The solid lines are for $\Phi$ fields, while the dashed lines represent
$\Sigma$ propagators.

  * •

$\dot{\Phi}\Sigma^{2}$ interaction: we first have the three-point contact
diagram with one $\phi$ and two $\chi$ fields (see Figure 3)

$\displaystyle\psi^{\prime}_{3}=i\alpha\int_{-\infty}^{\eta_{0}}d\eta
a(\eta)^{3}\partial_{\eta}K(k_{1},\eta)K(k_{2},\eta)K(k_{3},\eta)=\frac{\alpha}{H^{3}}\left[\frac{ik_{1}^{2}}{\eta_{0}}+k_{1}^{3}\log(-k_{t}\eta_{0})\right]\leavevmode\nobreak\
,$ (3.10)

which again contributes to a logarithmic IR divergent term in the
corresponding correlator. Next we also compute the four-point exchange diagram
with two $\phi$ and two $\chi$ fields as shown in Figure 3

$\displaystyle\psi^{\prime}_{4}$ $\displaystyle=$
$\displaystyle-\alpha^{2}\int_{-\infty}^{\eta_{0}}d\eta
d\eta^{\prime}a(\eta)^{3}a(\eta^{\prime})^{3}\partial_{\eta}K(k_{1},\eta)K(k_{2},\eta)G(s,\eta,\eta^{\prime})\partial_{\eta}K(k_{3},\eta)K(k_{4},\eta)$
(3.11) $\displaystyle=$
$\displaystyle-\frac{\alpha^{2}k_{1}^{3}k_{3}^{3}}{2H^{4}s^{3}}\left[\log\left(\frac{E_{L}}{k_{T}}\right)\log\left(\frac{E_{R}}{k_{T}}\right)+{\rm
Li}_{2}\left(\frac{k_{1}+k_{2}-s}{k_{T}}\right)+{\rm
Li}_{2}\left(\frac{k_{3}+k_{4}-s}{k_{T}}\right)\right.$
$\displaystyle\left.\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{2s(s^{2}-k_{3}(k_{3}+k_{4}))}{k_{3}(k_{3}+k_{4}-s)E_{R}}\log\left(\frac{E_{L}}{k_{T}}\right)+\frac{2s(s^{2}-k_{1}(k_{1}+k_{2}))}{k_{1}(k_{1}+k_{2}-s)E_{L}}\log\left(\frac{E_{R}}{k_{T}}\right)\right.$
$\displaystyle\left.\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{2s^{2}}{E_{L}E_{R}}\left(1+\frac{(k_{T}-k_{2})(k_{T}-k_{4})s}{k_{1}k_{3}k_{T}}+\frac{s^{2}}{k_{1}k_{3}}\right)-\frac{\pi^{2}}{6}\right]+\text{$t$-
and $u$-channels}.$

This wavefunction is actually IR-finite with no secular term depending on
$\eta_{0}$.666Naively it seems there are folded singularities at
$s=k_{1}+k_{2}$ and $s=k_{3}+k_{4}$, but one can easily check that they are
absent in the folded limit of the above result. A simple way to derive this is
to use weight-shifting operators and four-point scalar seed of massless
exchange as shown in Ref. [27]. Technically, it can be understood by looking
at the late-time limit of the bulk-to-bulk propagator in (3.5): $G$ decays as
$\eta^{3}$ thus the integrand in this case does not lead to significant
contribution in the late-time part of the bulk integration. However, the IR
behaviour of the four-point correlator $\langle\phi\chi\phi\chi\rangle$ is not
simply related to the $\psi_{4}$ above. As now the disconnected part with the
product of two ${\rm Re}\leavevmode\nobreak\ \psi_{3}$ gives $\log^{2}$
divergent terms, the IR-finite term from the connected part becomes sub-
dominant. Thus from this example, one can see the difference between
wavefunction coefficients and correlators. Especially their IR behaviours may
differ when we look at exchange diagrams. See [27] for more discussions about
the distinction.

There are several lessons that we can learn from the tree-level computation of
wavefunction coefficients. First, for interacting massless scalars,
logarithmic IR divergences arise when the number of derivatives is less than
2. For higher-derivative interactions one finds the $\psi_{n}$s are in term of
IR-finite rational polynomials [53]. Second, while the order of the divergence
is one for contact diagrams, it would depend on the type of interactions for
exchange diagrams. For the examples above, the exchange $\psi_{4}$ from two
$\Phi^{3}$ vertices has $\log^{2}$ divergence, since the super-horizon decay
of the $G$ propagator does not stop the growth of the integrand in (3.9).
Meanwhile the one from $\dot{\Phi}\Sigma^{2}$ is IR-finite because of the time
derivative in the vertex. But when we compute the corresponding correlators,
both exchange diagrams are IR-divergent, as the disconnected parts always lead
to $\log^{2}$-type contributions. Thus, in general we expect derivative
interactions lead to simpler IR-singular structure than the non-derivative
ones do. Technically this is because derivatives (either spatial or time ones)
increase the power of $\eta$ in the integrand and make it less likely to cause
late-time singular behaviour. Later we will show that this is also the case
for loop diagrams.

###  3.3 One-Loop Structure of IR-Divergent Correlators

Next we consider IR divergences in loop corrections to massless scalar
correlators using the wavefunction method. The analysis is rather technical
and it is better to gain some intuition from simple working examples. Thus
after a general discussion on loop integrals, we first focus on the one-loop
diagrams, and then look into higher-order loops in Section 3.4. Some of the
loop analysis have been sketched in Ref. [12], while in this paper we perform
a more explicit computation with concrete examples. All in all, the key
takeaway is a simplification: the infrared (IR) divergences in loop-level
correlation functions are consistently subdominant with respect dominant
classical loop contributions stemming from the tree-level wavefunction
coefficients. Consequently, the quantum contributions can be effectively
neglected in this context.

Let’s begin with a generic form of the loop wavefunction coefficients. By
using the Feynman rule we derived for quantum theories, we can quickly write
down the $L$-loop correction to a $n$-point wavefunction coefficient with $m$
vertices

$\displaystyle\psi_{n}^{L-\rm loop}$ $\displaystyle\sim$ $\displaystyle\int
d\eta_{1}...d\eta_{m}a(\eta_{1})^{4}...a(\eta_{m})^{4}K(k_{1},\eta_{1})...K(k_{n},\eta_{m})$
(3.12) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \int_{\bf
p_{1},...,p_{L}}G(p_{1},\eta_{a},\eta_{b})...G(p_{L},\eta_{c},\eta_{d})G(|{\bf
p_{x}+k_{y}}|,\eta_{e},\eta_{f})...$

where ${\bf p_{x}}$ and ${\bf k_{y}}$ represent various combinations of
internal and external momenta. One needs to be careful about the bulk time of
each insertion and the momentum conservation conditions for particular types
of interactions. Schematically we see that, in addition to the bulk time
integral for each vertex, there are also integrations for internal momenta.
These two types of integrations also arise in the in-in computation: the
former generates secular logarithmic terms (as for tree-level diagrams); while
the latter leads to IR divergences regularized by the cutoff $L$. In general
one would expect to see both types of divergences in the final correlator.
Although the computation looks similar in the wavefunction approach, it turns
out to be simpler for the above $\psi_{n}^{L-\rm loop}$.

To see this explicitly, let us briefly review an important observation in [12]
that, in the computation of loop-level wavefunction coefficients momentum
integrations do not become divergent at IR. We know that momentum integrals
can be regularized by the UV cutoff $\Lambda$ and the IR cutoff $1/L_{\rm
IR}$. UV divergences are usually renormalized by introducing counterterms. The
IR divergences are manifested in the $L_{\rm IR}\rightarrow\infty$ limit, and
we can use the following results of the loop integrations

$\displaystyle\int_{\bf p}\frac{1}{p^{n}}=\frac{1}{(2\pi)^{3}}\int_{1/L_{\rm
IR}}^{\Lambda}4\pi p^{2-n}dp\xrightarrow{L_{\rm
IR}\rightarrow\infty}\begin{dcases}\text{IR-finite}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ &n<3\\\ \frac{1}{2\pi^{2}}\log(kL_{\rm
IR})\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ &n=3\leavevmode\nobreak\
.\end{dcases}$

Then it is easy to see that all the momentum integrals in (3.12) are IR-finite
by noticing the soft behaviour of the bulk-to-bulk propagator

$\lim_{p\rightarrow
0}G(p,\eta,\eta^{\prime})=-\frac{i}{6}H^{2}(\eta^{3}+\eta^{\prime
3})+\mathcal{O}(p)\leavevmode\nobreak\ .$ (3.13)

Thus in the loop-level calculation of $\psi_{n}$ one can safely take the
$L_{\rm IR}\rightarrow\infty$ limit and the momentum integrals never become
singular at IR. As a result, the wavefunction coefficients can only have
secular IR divergences $\log(-k\eta_{0})$ from bulk time integration.

Next, we consider the computation of IR-divergent correlators using the
corresponding wavefunction coefficients. As explicit demonstrations, we derive
the two-point correlation functions of $\phi$ at one-loop level from three
different types of interactions in (3.6).

Figure 4: One-loop diagrams of two-point functions from three types of
interactions with IR divergences. The solid lines are for $\Phi$ fields, while
the dashed lines represent $\Sigma$ propagators.

  * •

One $\Phi^{4}$ vertex: This one-loop wavefunction coefficient has a single
internal line. Using the Feynman rules we derived in Section 2, we find

$\displaystyle\psi_{2}^{\prime\rm 1-loop}$ $\displaystyle=$
$\displaystyle\frac{i\lambda}{2}\int_{-\infty}^{\eta_{0}}d\eta
a(\eta)^{4}K(k,\eta)K(k,\eta)\int_{\bf p}G(p,\eta,\eta)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{6H^{2}}\int_{\bf
p}\left[\log(-2(k+p)\eta_{0})+\frac{k^{3}}{p^{3}}\log\left(\frac{k+p}{k}\right)+\gamma_{E}-\frac{i\pi}{2}-\frac{12k^{3}+6k^{2}p+13kp^{2}+28p^{3}}{12p^{2}(k+p)}\leavevmode\nobreak\
,\right]$

where there is a $1/2$ as the symmetric factor, and in the second line we
performed the time integral. As argued above, this loop correction to
$\psi_{2}$ has only secular divergence from bulk time integration. However, to
compute the loop correction to correlators from wavefunction coefficients, we
also need to take into account the classical loops from tree-level
wavefunction coefficients. For $\Phi^{4}$ interaction, this corresponds to the
second term in (2.26) which requires the $\psi_{4}$ in (3.7) from the contact
diagram. Then unlike the quantum loop from $\psi_{2}^{\rm 1-loop}$, the
momentum integral here leads to the IR divergence regularized by the cutoff
$L_{\rm IR}$. As a result, at $\lambda$ order the classical loop contribution
dominates over the quantum one, and the leading IR divergence in this
correlator is given by

$\langle\phi_{{\bf k}}\phi_{-{\bf k}}\rangle^{\prime}_{\rm
1-loop}\simeq-\frac{1}{8{\rm Re}\hskip 1.0pt\psi^{\prime}_{2}(k)\hskip
1.0pt{\rm Re}\hskip 1.0pt\psi^{\prime}_{2}(k)}\int_{\bf p}\frac{{\rm Re}\hskip
1.0pt\psi^{\prime}_{4}(k,k,p,p)}{{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(p)}=\frac{H^{2}}{2k^{3}}\frac{\lambda}{12\pi^{2}}\log(kL_{\rm
IR})\log(-2k\eta_{0})+...\leavevmode\nobreak\ $ (3.15)

This result agrees with the in-in computation as shown in Appendix A. From
this example, we see for the first time that the main contribution of the IR
divergence of a loop correlator is due to the classical wavefunction
coefficient.

  * •

Two $\Phi^{3}$ vertices: The one-loop correction here corresponds to the
second diagram in Figure 4, the expression of the wavefunction coefficient is
explicitly given in (2.18). Using the massless scalar propagators and
performing the time integral, we find

$\displaystyle\psi^{\prime\rm 1-loop}_{{\bf k}{\bf
k}^{\prime}}=-\frac{g^{2}}{27H^{4}}\int_{\bf
p}\log\left(-(2k+p)\eta_{0}\right)\leavevmode\nobreak\ ,$ (3.16)

which again only contains secular-type IR divergence, as the loop integration
is IR-finite. To compute the two-point correlator at one-loop ($g^{2}$) order,
we need to use the relation (2.26) derived by the Born rule, and all three
terms there are present for the $\Phi^{3}$ theory. As the first term with
$\psi^{\rm 1-loop}_{2}$ has no loop IR divergences, the leading contributions
come from the second and third terms, which are explicitly given by

2nd term $\displaystyle=$
$\displaystyle\left(\frac{H^{2}}{2k^{3}}\right)^{2}\int_{\bf
p}\frac{H^{2}}{2p^{3}}{\rm Re}\hskip 1.0pt\psi_{4}^{\prime}({{\bf k},{\bf
k}^{\prime}},{\bf p},-{\bf
p})=\frac{g^{2}}{8k^{3}}\frac{-5}{18\pi^{2}}\log(kL_{\rm
IR})\log(-2k\eta_{0})^{2}\leavevmode\nobreak\ ,$ 3rd term $\displaystyle=$
$\displaystyle\left(\frac{H^{2}}{2k^{3}}\right)^{2}\int_{\bf
p}\frac{H^{2}}{2p^{3}}\left[\frac{H^{2}}{2|{\bf k}+{\bf p}|^{3}}2{\rm
Re}\hskip 1.0pt\psi^{\prime}_{3}({\bf k},{\bf p},-{\bf p}-{\bf k}){\rm
Re}\hskip 1.0pt\psi^{\prime}_{3}({\bf k}^{\prime},-{\bf p},{\bf p}+{\bf
k})\right.$ (3.18) $\displaystyle\left.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +\frac{H^{2}}{2|{\bf k}+{\bf
k}^{\prime}|^{3}}2{\rm Re}\hskip 1.0pt\psi^{\prime}_{3}({\bf k},{\bf
k}^{\prime},-{\bf k}-{\bf k}^{\prime}){\rm Re}\hskip
1.0pt\psi^{\prime}_{3}({\bf k}+{\bf k}^{\prime},{\bf p},-{\bf p})\right]$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{8k^{3}}\frac{1}{3\pi^{2}}\log(kL_{\rm
IR})\log(-2k\eta_{0})^{2}\leavevmode\nobreak\ ,$

where $\psi_{3}$ and $\psi_{4}$ are the tree-level wavefunction coefficients
(3.8) and (3.9) from contact and exchange diagrams. As both of them have
$\log^{2}$ order secular divergences, the two-point correlator in the end is a
sum of two contributions

$\langle\phi_{{\bf k}}\phi_{-{\bf k}}\rangle^{\prime}_{\rm
1-loop}\simeq\frac{g^{2}}{8k^{3}}\frac{1}{18\pi^{2}}\log(kL_{\rm
IR})\log(-2k\eta_{0})^{2}\leavevmode\nobreak\ ,$ (3.19)

which again reproduces the in-in result shown in Appendix A. This example also
shows that the leading IR divergence of the correlator is given by classical
loops from tree-level wavefunction coefficients, although in this case both
the contact $\psi_{3}$ and exchange $\psi_{4}$ contribute at the same order.

  * •

Two $\dot{\Phi}\Sigma^{2}$ vertices: Here the one-loop process also
corresponds to the third diagram in Figure 4, though the two internal lines
are for $\Sigma$ field and there is one time derivative on $\Phi$ for each
vertex

$\psi^{\prime\rm 1-loop}_{{\bf k}{\bf k}^{\prime}}=-\frac{\alpha^{2}}{2}\int
d\eta
d\eta^{\prime}a(\eta)^{3}a(\eta^{\prime})^{3}\partial_{\eta}K(k,\eta)\partial_{\eta^{\prime}}K(k,\eta^{\prime})\int_{\bf
p}G(p,\eta,\eta^{\prime})G(|{\bf k}+{\bf
p}|,\eta,\eta^{\prime})\leavevmode\nobreak\ .$ (3.20)

Then because of the time derivative, this wavefunction coefficient is actually
IR-finite, even without secular logarithms from time integrations, as we found
for the exchange $\psi_{4}$ in (3.11). However, since the contact $\psi_{3}$
in (3.10) has a secular term, we still get IR-divergent contributions from the
third term in (2.26).

$\displaystyle\langle\phi_{{\bf k}}\phi_{-{\bf k}}\rangle^{\prime}_{\rm
1-loop}$ $\displaystyle\simeq$ $\displaystyle\frac{1}{8{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(k)\hskip 1.0pt{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(k)}\int_{\bf p}\frac{{\rm Re}\hskip
1.0pt\psi^{\prime}_{3}({\bf k},{\bf p},-{\bf p}-{\bf k}){\rm Re}\hskip
1.0pt\psi^{\prime}_{3}(-{\bf k},-{\bf p},{\bf p}+{\bf k})}{{\rm Re}\hskip
1.0pt\psi^{\prime}_{2}(p){\rm Re}\hskip 1.0pt\psi^{\prime}_{2}(|{\bf p}+{\bf
k}|)}$ (3.21) $\displaystyle=$
$\displaystyle\frac{H^{2}}{2k^{3}}\frac{\alpha^{2}}{8\pi^{2}}\log(kL_{\rm
IR})\log(-2k\eta_{0})^{2}\leavevmode\nobreak\ ,$

which also reproduces the in-in result in Appendix A. Again, we find the
correlator is dominated by the contribution from classical loops. This time
the computation is further simplified, as the exchange wavefunction
coefficient does not contribute and one only needs to consider the term from
the product of two contact $\psi_{3}$.

#### Summary

From these three examples, we see that leading IR divergences of correlators
come from classical loops, which means only tree-level wavefunction
coefficients are needed. While the explicit computations above are for two-
point functions, it is straightforward to extend the analysis to any higher-
point functions with IR-divergent one-loop corrections. In general, the
momentum integrations in loop-level wavefunction coefficients do not lead to
IR divergences, and thus the quantum loops only contain secular divergences,
as we see below (3.12). Meanwhile, at the same order of the couplings, the
classical loops contain both IR divergences from momentum integrals and also
secular terms from time integrals, thus giving rise to the leading logarithms
in correlators. In general, to explicitly compute the classical loops for
higher-point functions, we need wavefunction coefficients of more complicated
tree-level exchange diagrams with multiple internal lines. The integration
over the internal momentum $\bf p$ is IR-divergent as there is always a
$\psi_{2}(p)\propto p^{3}$ in the denominator of the integrand. As a result,
the classical loops always give the dominant contribution to the final
correlators.

###  3.4 Multi-Loop Diagrams and Leading Logarithms

Although the explicit computation becomes complicated once we consider multi-
loop processes, the notion of classical loops can still help us easily
identify the leading logarithms in IR-divergent correlators. Let’s take
$g\Phi^{3}$ interaction as an explicit example. At the $g^{V}$ order, the
$n$-point correlator of $L$-loop diagram can be computed by using all the
wavefunction coefficients and their products with $V$ vertices. Schematically,
the relation is

$\displaystyle\langle\phi^{n}\rangle_{L-\rm loop}$
$\displaystyle\sim\frac{1}{({\rm Re}\psi_{2})^{n}}\Bigg{[}{{\rm
Re}\leavevmode\nobreak\ \psi_{n}^{L-\rm loop}}+\int_{\bf p_{1}}\frac{{\rm
Re}\psi_{n+2}^{(L-1)-\rm loop}}{{\rm Re}\psi_{2}(p_{1})}+...+\int_{\bf
p_{1},...,p_{L-1}}\frac{{\rm Re}\psi_{n+2(L-1)}^{1-\rm loop}}{({\rm
Re}\psi_{2}(p_{1})...{\rm Re}\psi_{2}(p_{L-1}))}$ $\displaystyle+\int_{\bf
p_{1},...,p_{L}}\frac{1}{{\rm Re}\psi_{2}(p_{1})...{\rm
Re}\psi_{2}(p_{L})}\left({\rm Re}\psi^{\rm ex}_{n+2L}+\frac{{\rm Re}\psi^{\rm
ex}_{n+2L-1}{\rm Re}\leavevmode\nobreak\ \psi_{3}}{{\rm
Re}\leavevmode\nobreak\ \psi_{2}}+...+\frac{({\rm Re}\leavevmode\nobreak\
\psi_{3})^{V}}{({\rm Re}\leavevmode\nobreak\
\psi_{2})^{3V-2L-n}}\right)\Bigg{]}\leavevmode\nobreak\ ,$ (3.22)

where all the numerical coefficients are dropped. In the first line we collect
all the wavefunction coefficients containing loops, and the tree-level
contributions are listed in the second line, which are the classical loops.
One example is shown in Figure 5. As we have argued in the one-loop case, the
momentum integration in wavefunction coefficients does not lead to IR
divergences, while each momentum integration explicitly shown in (3.4)
contributes one $\log(k_{E}L_{\rm IR})$.777Here $k_{E}$ represents the
magnitude for one combination of various external momenta. The same notation
applies for the discussion in the rest of the section. Thus the second line
leads to more divergent contributions than the ones from loop-level
wavefunction coefficient. In other words, the classical loops always dominate
over the quantum ones for IR-divergent correlators.

Figure 5: A diagrammatic illustration of a multi-loop relation: the correlator
on the left (the diagram with a ragged black line as the late-time boundary)
receives various contributions from both loop-level (the first and second
lines) and tree-level (the third line) wavefunction coefficients on the right
(the diagrams with grey late-time boundaries). In this example one needs to
incorporate all the $\psi_{n}$’s and their products at the $g^{7}$ order, as
required by the Born rule (2.20) in perturbation theory.

This analysis in the end leads us to a simple conclusion: the saddle-point
approximation of the wave function (2.6), which contains all the tree-level
information, captures the leading IR behaviour. Intuitively, it is easy to
understand: IR divergences are generated on superhorizon scales, which are
normally due to semi-classical effects. As we have shown in Section 2, the
wavefunction of the Universe provides a clear distinction for quantum and
classical effects, which helps us here to confirm the intuitive understanding
with concrete computations. This interesting conclusion provides
simplifications for the analysis of IR divergences in dS, and will play an
important role for our discussion of the stochastic formalism in the following
sections. The same results can be found for the $\lambda\Phi^{4}$ and
$\alpha\dot{\Phi}\Sigma^{2}$ interactions.

As one example of the simplifications, now let us show that in the
perturbative computation how we can directly identify the leading IR
logarithms. The trick here is to simply look at the last term in the second
line in (3.4), which is the contribution from the product of $V$ three-point
wavefunction coefficient. As contact diagrams are simple, the IR divergence of
${\rm Re}\psi_{3}$ is known in (3.8), with each of them contributing a secular
term $\log(-k_{\Sigma}\eta_{0})$. Meanwhile, each classical loop integral
contains a $\log(k_{E}L_{\rm IR})$. As a result, we find the leading
logarithmic divergence for this $L$-loop $n$-point correlator is given by

$\langle\phi^{n}\rangle_{L-\rm loop}\propto g^{V}\log(k_{E}L_{\rm
IR})^{L}\log(-k_{\Sigma}\eta_{0})^{V}\leavevmode\nobreak\ .$ (3.23)

One can quickly check that the above result also works for other interactions
with IR divergences. In general, considering an $m$-point vertex, we have the
relation $L=1+\left(\frac{m}{2}-1\right)V-\frac{n}{2}$ for diagrams with $n$
external legs, $V$ vertices and $L$ loops. Then for the two-point functions
from a given interaction, the order of leading divergences depends on the
number of vertices only

$\displaystyle\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{L-\rm
loop}\propto\begin{cases}\lambda^{V}\log(k_{E}L_{\rm
IR})^{V}\log(-k_{\Sigma}\eta_{0})^{V}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ &{\rm for}\leavevmode\nobreak\ \Phi^{4}\\\
g^{V}\log(k_{E}L_{\rm
IR})^{V/2}\log(-k_{\Sigma}\eta_{0})^{V}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ &{\rm for}\leavevmode\nobreak\
\Phi^{3}\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\
\dot{\Phi}\Sigma^{2}\leavevmode\nobreak\ .\end{cases}$ (3.24)

Thus the order of divergences and the power of the couplings are uniquely
related for classical loops. This fact plays an important role in the
following analysis about resummation. Meanwhile the IR divergences from
quantum loops for the same diagram are suppressed, as they have the same power
of the coupling, but lower power of logarithmic functions. In the next section
we will show that when we go to the coordinate space, this leading-log
behaviour agrees with the result in Ref. [13] by using the retarded in-in
formalism.

Finally, let’s comment on the differences for the various interactions. For
$g\Phi^{3}$ and $\lambda\Phi^{4}$, the leading divergent piece may also
receive classical loop contributions from wavefunction coefficients of
exchange diagrams, which have the same order of $\log$s with the products of
contact $\psi_{n}$s, as we have seen in the previous analysis of the one-loop
correlator from $\Phi^{3}$ interaction. The situation for
$\dot{\Phi}\Sigma^{2}$ can be further simplified, since the only IR-divergent
tree-level wavefunction coefficient is the contact $\psi_{3}$ in (3.10). As a
result, the last term with the product of ${\rm Re}\psi_{3}$s in (3.4) gives
us all the contribution to the leading logarithms of IR-divergent correlators.
This means for this special two-field theory, all the higher-order $\psi_{n}$s
in the semi-classical wavefunction are IR-safe, and we only need the simplest
tree-level term $\psi_{3}$ to capture its IR-singular behaviour. This simple
toy model example provides an interesting playground for studying IR
divergences, which even allows us to go beyond perturbation theory as we shall
discuss in Appendix B.

##  4 What does the Stochastic Formalism Resum?

The stochastic formalism provides another framework to study IR effects of
quantum field theories in dS, which is supposed to work even in the non-
perturbative regime. In this approach, the Fokker-Planck equation is applied
to study the probability distribution $P$ of long wavelength perturbations. It
has been argued for a long time that by doing so, there is an equilibrium
state which resums the IR-divergences arising from all the higher loop
contributions, and thus in the end no IR divergences remain for interacting
massless scalars in dS. However, it is still unclear how this resummation was
achieved, and how this formalism connects with the perturbation theory.
Meanwhile, for any quantum system, it is well-known that the probability
distribution is simply given by $P=|\Psi|^{2}$. Therefore the wavefunction of
the Universe provides an obviously natural setup to look into this remaining
issue. With results of IR-divergent correlators from field-theoretic
computations in perturbation theory, now we explore the connection with the
stochastic formalism.

In this section, we work in the perturbative regime, and explicitly compare
the calculation of IR-divergent correlators from the coarse-grained
wavefunction (Section 4.1) with the results from the stochastic formalism
(Section 4.2). Interestingly, this analysis in the end provides a simple
answer to the question proposed in the title of the section:

The stochastic formalism is a resummation of classical loops.

In addition to comparing perturbative correlators in this section, we shall
provide more supportive evidence in Appendix B by computing the probability
distribution directly from the wavefunction. Then in Section 5 we further
justify this conclusion in the non-perturbative regime.

###  4.1 Long Modes Correlations from the Wavefunction

To make the comparison, there is one more step for the computation of
correlators in the wavefunction approach. Since the stochastic formalism
describes long wavelength perturbations, we need to coarse grain the system by
integrating over the short wavelength modes. Let’s first define the long
wavelength perturbations in coordinate space by using a window function

$\phi_{l}({\bf x})=\int_{\bf
k}\Omega_{\Lambda}(k)e^{i\mathbf{k}\cdot\mathbf{x}}\phi_{\bf
k}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \chi_{l}({\bf x})=\int_{\bf
k}\Omega_{\Lambda}(k)e^{i\mathbf{k}\cdot\mathbf{x}}\chi_{\bf k}.$ (4.1)

This window function $\Omega_{\Lambda}(k)$ vanishes for $k>\Lambda$ and
approaches $1$ for $k<\Lambda$, with $\Lambda$ being the UV cutoff. One simple
example of the window function is the Heaviside step
$\Omega_{\Lambda}(k)=\Theta(\Lambda-k)$, and there are also smooth versions of
the window function which preserve the locality of short-scale interactions
[12]. The role of the UV cutoff is to define a fixed physical length
$L_{UV}=a(t)/\Lambda$ such that perturbations with shorter wavelengths are
smoothed out. This means the comoving scale $\Lambda$ should be time-
dependent. Meanwhile, as argued in [79], the minimum length scale for
smoothing should be larger than the Hubble radius such that the quantum nature
of fields can be neglected. We choose $\Lambda(t)=\epsilon a(t)H$ with
$\epsilon\ll 1$. As this UV cutoff is artificial, it should not appear in
physical observables at the end of the computation.

With this window function, we will look into the coarse-grained wavefunction
in Section 5, and derive the probability distribution of the long wavelength
perturbations in Appendix B. Here in this section we compute correlators of
long wavelength modes using the results of perturbative wavefunction. In order
to compare with the stochastic formalism, we are interested in the correlators
at the coincident point ${\bf x}$, which can be expressed in terms of the
following Fourier transformation

$\displaystyle\langle\phi_{l}({\bf x})^{n}\rangle\equiv\int_{{\bf
k}_{1},\dots,{\bf
k}_{n}}\prod_{i=1}^{n}{\Omega_{\Lambda}(k_{i})}\langle\phi_{{\bf
k}_{1}}\dots\phi_{{\bf k}_{n}}\rangle e^{i({\bf k}_{1}+\dots+{\bf
k}_{n})\cdot{\bf x}}\leavevmode\nobreak\ ,$ (4.2)

where $\langle\phi_{{\bf k}_{1}}\dots\phi_{{\bf k}_{n}}\rangle$ are the
correlators computed in Section 3.

Let’s begin with the free theory in order to explain our choice for
regularization. In this simplest case, the wavefunction is fully determined by
$\psi_{2}$ in (3.3), and the two-point function at ${\bf x}=0$ is just the
variance of Gaussian distribution

$\sigma^{2}\equiv\langle\phi_{l}({\bf x}=0)^{2}\rangle=\int_{\bf
k}\frac{\Omega_{\Lambda}(k)}{2{\rm
Re}\psi_{2}(k)}=\frac{H^{2}}{4\pi^{2}}\int_{1/L_{\rm
IR}}^{\Lambda}\frac{dk}{k}=\frac{H^{2}}{4\pi^{2}}\log(\Lambda L_{\rm
IR})\leavevmode\nobreak\ .$ (4.3)

Naively, this two-point correlator in real space seems to depend on both the
IR and UV cutoff scales, which should not be the case for physical
observables. However, the important part of the above logarithmic function is
its time-dependence, while the rest can be seen as regularization-dependent
constants. Recall that the comoving size of the Universe can be associated
with some initial time $L_{\rm IR}^{-1}=a(t_{i})H$, and we are free to pick a
scale factor normalization $a(t_{i})=\epsilon$. As a result, we get

$\langle\phi_{l}^{2}\rangle=\frac{H^{2}}{4\pi^{2}}\log
a(t)\leavevmode\nobreak\ ,$ (4.4)

which reproduces the famous result for the stochastic behaviour of a free
massless scalar in de Sitter, $\sigma^{2}=H^{3}t/(4\pi^{2})$ by using
$a(t)=e^{Ht}$. Physically, this secular growth with time can be explained as
follows: the physical size of the box is expanding with $a(t)L_{\rm IR}$ while
the smallest scale for screening is fixed to be
$L_{UV}=a(t)/\Lambda(t)=1/(\epsilon H)$. Therefore, there are more and more
comoving modes entering the coarse-grained system that we are looking at,
while no long wavelength perturbations are leaving the box.

Higher order correlation functions will follow the same logic, ie. when
integrating over momenta we will consider the same integration limits. Notice
that the presence of IR divergence translates into higher powers of $\log a$.
For instance, whereas the four point function in $\lambda\phi^{4}$ contains a
single power of $\log(-k_{T}\eta_{0})$ the correlator
$\langle\phi_{l}^{4}\rangle$ goes as $\lambda H^{4}\log a^{4}$. The reason
behind this is because there are now four different momenta entering the
horizon, each adding a factor of $\log a$. Notice that the number of IR
divergences is crucial, as it clearly indicates the larger terms. In this
section we will make use of this fact to understand how the semi-classical
solution is always the dominant one.

We can make more explicit the relation between perturbative computations and
the stochastic formalism. To do so let us start by considering a collection of
fields $\phi^{a}$ where $a$ labels the field. The probability distribution
functions for the long wavelength part of $\phi^{a}_{l}$ can be written as,

$\displaystyle P[\phi_{l}^{a}]=\int\left[\prod_{a}\mathcal{D}\phi_{\bf k}^{a}\
\delta\left(\phi_{l}^{a}-\int_{\mathbf{k}}\Omega_{\Lambda}(k)\phi_{\bf
k}^{a}\right)\right]|\Psi[\phi_{\bf k}^{a}]|^{2}\leavevmode\nobreak\ .$ (4.5)

where $\Omega_{\Lambda}(k)$ is the window function defined earlier. As has
been proven for a single scalar field in [12], the time evolution of
$P[\phi_{l}^{a}]$ is given by the Fokker-Planck equation (We show how this is
extended to two fields in Appendix C). If we write the wavefunction
$\Psi[\phi^{a}]$ on perturbation theory, the path integral will contains all
long wavelength correlation functions. To see this fact, in detail it is
instructive to write the $\delta$ function in (B.1) as an exponential,

$\displaystyle P[\phi_{l}^{a},t]=\int\mathcal{D}\phi_{k}^{a}\int
dJ^{a}\exp\left[i\sum_{a}\left(\phi_{l}^{a}-\int_{\mathbf{k}}\Omega_{k}\phi_{k}^{a}\right)J^{a}\right]|\Psi(\phi_{k}^{a})|^{2}\leavevmode\nobreak\
.$ (4.6)

Upon closer examination, the path integral over the short modes can be viewed
as two successive Legendre transforms. The first transform introduces a
current that only has support on the long-wavelength modes, while the second
transform focuses on the long-wavelength part of the fields.

Let’s delve into the first transformation. By introducing a current with
restricted support over the square of the wavefunction, we effectively
generate all connected correlators. This process is analogous to the creation
of a linear interaction term, $\phi J$, in quantum field theory. After
integrating over $\phi_{\bf k}$, we obtain all correlators with an arbitrary
number of external $J$ currents. The number of external legs, denoted as $J$,
encompasses all possible ways of connecting these legs using the available
wavefunction coefficients. This includes not only tree-level diagrams but also
all loops (up to the order specified by the wavefunction coefficients). This
object corresponds to a partition function for the long wavelength
correlators, denoted as $Z[J]$, whose expression is left in Appendix B. As
usual, we can compute correlation functions by taking functional derivatives
of $\ln Z[J]$ and setting $J$ to 0. This operation effectively eliminates all
external legs, leaving us with the connected correlations. In this sense the
partition function computes the same quantities as the perturbative solution
to the Fokker-Planck equation. Moreover, one of the advantages of solving the
path integral is that we can understand how the semi-classical part dominates
over all quantum corrections.

Next, we use this approach to derive the long modes correlators from the
perturbative wavefunction. Our analysis again covers three types of
interactions shown in (3.6).

  * •

$\lambda\Phi^{4}/4!$ interaction: Let us start by considering the quartic
vertex. Either using the partition function or the relation (4.2) , we can
write down the two-point function of long modes as

$\displaystyle\langle\phi^{2}\rangle$
$\displaystyle\equiv\left.\frac{\delta^{2}\log Z[J]}{\delta
J^{2}}\right|_{J=0}$
$\displaystyle=\frac{1}{Z[0]}\int_{\mathbf{k}}\frac{\Omega(k)}{2\mathrm{Re}\psi_{2}(\mathbf{k})}\left[1+\int\mathcal{D}\phi_{k}|\Psi|^{2}\left(\int_{\bf
p}\frac{2\mathrm{Re}\psi_{4}(\mathbf{k},-\mathbf{k},\mathbf{p},-\mathbf{p})}{2\mathrm{Re}\psi_{2}(\mathbf{p})}\right)\phi_{k}^{2}+\dots\right]\leavevmode\nobreak\
,$ (4.7)

where dots represent terms higher order in $\lambda$. Here we focus solely on
the tree-level wavefunction coefficients, and the contribution from the first
term alone yields the conventional two-point function in the free theory. The
window function $\Omega_{\Lambda}(k)$ effectively eliminates all short-
wavelength modes. Then the second term with the tree-level wavefunction
coefficient $\psi_{4}$ gives the leading one-loop correction to the
correlation function, as we have found in (3.15). Performing the $\bf k$
integration of the smeared Fourier transformation, we find

$\displaystyle\langle\phi_{l}^{2}({\bf x}=0)\rangle$
$\displaystyle=\int_{\mathbf{k}}\Omega(k)\frac{H^{2}}{2k^{3}}\left[1+\frac{\lambda}{12\pi^{2}}\log(kL_{\mathrm{IR}})\log(-2k\eta_{0})+\mathcal{O}(g^{4})\right]$
$\displaystyle=\frac{H^{2}}{4\pi^{2}}\log a-\frac{\lambda
H^{2}}{144\pi^{2}}(\log a)^{3}+\mathcal{O}(\lambda^{2}(\log
a)^{5})\leavevmode\nobreak\ .$ (4.8)

The leading logarithmic divergences from two-loop and arbitrarily higher-loop
terms can be estimated in a similar way, though the numerical coefficients may
become very difficult to compute. For the $\Phi^{4}$ theory, the $L$-loop
correction to the power spectrum from classical loops is given in (3.24),
which lead to the following correction to the two-point function in coordinate
space

$\langle\phi_{l}^{2}\rangle_{L-{\rm loop}}\propto\lambda^{V}(\log
a)^{2V+1}\leavevmode\nobreak\ ,$ (4.9)

where the number of vertices satisfies $V=L$ for the two-point function from
quartic interactions. This result of the leading logarithms is in agreement
with the analysis in Ref. [13].888To make the explicit comparison, we notice
that the number of internal lines for a Feynman diagram satisfies $I=L+V-1$.
In Ref. [13], the leading order is given by $\lambda^{V}\log^{P}a$, with
$P=n+I$ being the total number of propagators. For the two-point function in
$\Phi^{4}$ theory, we simply have $P=2V+1$. Basically, it shows that if we go
to higher order in perturbation theory, the corrections come as a power of the
combination $\lambda(\log a)^{2}$.

Let us now consider the corrections from quantum loops. At the linear order in
$\lambda$, we find a contribution from $\psi_{2}^{1-\mathrm{loop}}$ in (• ‣
3.3). After integrating over the external momentum $\bf k$, we find a term
proportional to $\lambda\log^{2}a$. This contribution is clearly subdominant
compared to the loop correction from the tree-level wavefunction coefficient.
To isolate the semi-classical contributions, we may define an effective
coupling $\tilde{\lambda}=\lambda(\log a)^{2}$ and consider the limit
$\lambda\to 0$ and $\log a\to\infty$ with $\tilde{\lambda}$ being constant.
Then the quantum loops are suppressed by a factor of $\tilde{\lambda}/\log a$
at least. This analysis can be simply extended to higher order terms.

  * •

$g\Phi^{3}/3!$ interaction: The cubic vertex follows the same logic as the
quartic case. After a straightforward computation the two-point function is
given by

$\displaystyle\langle\phi_{l}^{2}\rangle$
$\displaystyle\equiv\left.\frac{\delta^{2}\log Z[J]}{\delta
J^{2}}\right|_{J=0}$
$\displaystyle=\frac{1}{Z[0]}\int_{\mathbf{k}}\frac{\Omega(k)}{2\mathrm{Re}\psi_{2}(\mathbf{k})}$
$\displaystyle\times\left[1+\int\mathcal{D}\phi_{k}|\Psi|^{2}\left(\int_{\mathbf{p}}\frac{\mathrm{Re}\psi_{4}(\mathbf{k},\mathbf{k}^{\prime},\mathbf{p},-\mathbf{p})}{2\mathrm{Re}\psi_{2}(\mathbf{p})}\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{\mathbf{p}}\frac{2\mathrm{Re}\psi_{3}(\mathbf{k},\mathbf{p},-\mathbf{p},-\mathbf{k})\mathrm{Re}\psi_{3}(\mathbf{k}^{\prime},\mathbf{p},-\mathbf{p},\mathbf{p+k})}{2\mathrm{Re}\psi_{2}(\mathbf{p})2\mathrm{Re}\psi_{2}(|\mathbf{k}+\mathbf{p}|)}\right)\phi_{\mathbf{k}}^{2}+\dots\right]\leavevmode\nobreak\
.$ (4.10)

Within the parentheses, the path integral in the second term corresponds to a
1-loop correction arising from the exchange diagram $\psi_{4}$ as well as from
the product of two $\psi_{3}$ functions, as derived in (3.19). Consequently,
the expression becomes

$\displaystyle\langle\phi_{l}^{2}({\bf x}=0)\rangle$
$\displaystyle=\int_{\mathbf{k}}\Omega(k)\frac{H^{2}}{2k^{3}}\left[1+\frac{g^{2}}{72\pi^{2}}\log(kL_{\mathrm{IR}})\log(-2k\eta_{0})^{2}+\mathcal{O}(g^{4})\right]$
$\displaystyle=\frac{H^{3}}{4\pi^{2}}\log
a\left[1+\frac{g^{2}}{288\pi^{2}}(\log a)^{3}+\mathcal{O}(g^{4}(\log
a)^{6})\right]\leavevmode\nobreak\ .$ (4.11)

Similarly, we can also include the quantum loop corrections to this
correlator. At the leading ($g^{2}$) order, we find a contribution from
$\psi_{2}^{1-\mathrm{loop}}$ in (3.16) which results in a term proportional to
$g^{2}\log a^{3}$ and is subdominant compared to the classical loops. To
isolate the semi-classical contributions, again we may introduce the coupling
$\tilde{g}=g\log a^{3}$ and consider the limit as $g\to 0$, $\log a\to\infty$,
while $\tilde{g}$ remains finite. For instance, in the given example, the
quantum correction from $\psi_{2}^{1-\mathrm{loop}}$ is suppressed by a factor
of $\tilde{g}/\log a$.

Focusing on classical loop contributions, we can simply write down higher
order corrections from $L$-loop processes by using the leading logarithmic
divergences that we identified in (3.24). For two-point correlators of long
modes in coordinate space, we find

$\langle\phi_{l}^{2}\rangle_{L-{\rm loop}}\propto g^{2L}(\log
a)^{3L+1}\leavevmode\nobreak\ ,$ (4.12)

where we have used the relation for the number of vertices $V=2L$ for two-
point functions from cubic interactions. This tells us at each order in
perturbation theory, the leading contributions to IR divergences come with the
combination $g^{2}\log^{3}a$. Quantum loops only lead to subdominant
contributions suppressed by lower powers of $\log a$.

  * •

$\alpha\dot{\Phi}\Sigma^{2}/2$ interaction: This two-field case is actually
simpler, as in the semi-classical wavefunction only the contact $\psi_{3}$ in
(3.10) contains secular growth term. Therefore, we are allowed to neglect
$\psi_{n>3}$ from exchange diagrams, and the semi-classical wavefunction with
$\psi_{2}$ and $\psi_{3}$ captures the leading IR divergences in this theory.
Furthermore, we can explicitly perform the integration over the fields and
obtain the following generator999We leave further details of this computation
in Appendix B. There, we also present the computation for the probability
distribution and confirm that the result from this simplified semi-classical
wavefunction is a solution of the Fokker-Planck equation. This provides
another support for our conclusion that the stochastic formalism is resumming
classical loops.

$\displaystyle
Z[J_{\phi},J_{\chi}]=\exp\left\\{-\frac{\langle\phi^{2}\rangle}{2}J_{\phi}^{2}+\frac{\langle\chi^{2}\rangle}{1-i\frac{\langle\phi\chi^{2}\rangle}{\langle\phi^{2}\rangle}J_{\phi}}\frac{J_{\chi}^{2}}{2}\right\\}\leavevmode\nobreak\
.$ (4.13)

It’s worth noting that the cubic interaction introduces a non-local term,
which corresponds to a resummation of diagrams with two legs of $\chi$ and an
arbitrary number of legs of $\phi$. It can be readily verified that the two-
point function is given by

$\displaystyle\langle\phi^{2}\rangle=\frac{H^{2}}{4\pi^{2}}\log
a+\frac{\alpha^{2}H^{2}}{16\pi^{2}}(\log a)^{4}\leavevmode\nobreak\ .$ (4.14)

Notice that integrating over $J_{\chi}$ yields a collection of classical one-
loop diagrams with an arbitrary number of external $\phi$ legs. This result is
similar to the one obtained in [80] using a different technique but with a
similar coupling. The consequences of this will be explored elsewhere.

For this interaction, the contributions from quantum loops are IR-finite and
thus absent, as we have seen in (3.20). Meanwhile, the leading logarithms for
arbitrary higher order loop diagrams have the same behaviour as in (4.12), as
the classical loops here are similar with the ones from the $\Phi^{3}$
interaction.

This concludes our perturbative analysis using the wavefunction method. After
identifying the leading IR divergences from classical loops, now we are ready
to make the comparison with perturbative computations of the stochastic
formalism.

###  4.2 Correlators in the Perturbative Stochastic Formalism

Now we consider the stochastic formalism and try to establish the link with
the field-theoretic computations. This approach is supposed to be an effective
description for the stochastic behaviour on superhorizon scales while the
short wavelength perturbations are smoothed out. In order to make the explicit
comparison with the wavefunction calculation of three types of interactions in
(3.6), here we look at the following two-field system for the stochastic
analysis

$\mathcal{L}_{l}=\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\dot{\chi}^{2}-f(\chi)\dot{\phi}-V(\phi,\chi)\leavevmode\nobreak\
.$ (4.15)

where the gradients of fields have been neglected. Strictly speaking, the two
scalars here $\phi(t)$ and $\chi(t)$ correspond to boundary fluctuations
$\phi_{l}$ and $\chi_{l}$ in the field-theoretic approach. Then (4.15) can be
seen as an approximate Lagrangian around the future boundary with time $t$
where we perform the bootstrap analysis. We keep the time dependence of these
long wavelength modes as the system may still evolve. The three types of
interactions can be achieved by choosing different forms of the potential
function $V(\phi,\chi)$ and the $f(\chi)$ function for the kinetic coupling.

The starting point of the stochastic approach is the Fokker-Planck equation
for the probability distribution of long wavelength perturbations. Here we
consider an extended version of the Fokker-Planck equation that also works for
the long-wavelength system in (4.15)

$\displaystyle\frac{dP}{dt}=\frac{\partial}{\partial\phi}\left[\left(\frac{V_{\phi}}{3H}+f(\chi)\right)P\right]+\frac{\partial}{\partial\chi}\left[\left(\frac{V_{\chi}}{3H}\right)P\right]+\frac{H^{3}}{8\pi^{2}}\left(\frac{\partial^{2}P}{\partial\phi^{2}}+\frac{\partial^{2}P}{\partial\chi^{2}}\right)\leavevmode\nobreak\
.$ (4.16)

We leave the details of its derivation in Appendix C, and here focus on its
implications for correlators. As is well-known, the Fokker-Planck equation
describes the time evolution of the probability distribution $P[\phi,\chi;t]$
with two physical effects: drift, which is given by the first two terms on the
right-hand side of (4.16) above and comes from the background equation of
motion; diffusion, which is the last term caused by quantum noise. In the non-
perturbative regime, these two effects are supposed to compete with each other
and in the end lead to an equilibrium state with ${dP}/{dt}=0$.

With a given probability distribution function, correlation functions can be
computed using the Born rule:

$\displaystyle\langle\phi^{n}\chi^{m}\rangle=\int\mathcal{D}\phi\mathcal{D}\chi\phi^{n}\chi^{m}P[\phi,\chi;t]\leavevmode\nobreak\
.$ (4.17)

By substituting (4.17) into the Fokker-Planck equation, we derive a system of
differential equations governing the evolution behavior of the correlation
functions. Assuming that $V(\phi,\chi)$ are polynomial in the fields, we can
write these equations into the following form

$\displaystyle\frac{d}{dt}\langle\phi^{n}\chi^{m}\rangle$ $\displaystyle=\int
d\phi d\chi\phi^{n}\chi^{m}\frac{P(\phi,\chi,t)}{dt}$
$\displaystyle=-\frac{n}{3H}\left\langle\phi^{n-1}V_{\phi}\chi^{m}\right\rangle-n\langle\phi^{n-1}\chi^{m}f(\chi)\rangle-\frac{m}{3H}\left\langle\phi^{n}\chi^{m-1}V_{\chi}\right\rangle$
$\displaystyle+\frac{n(n-1)H^{3}}{8\pi^{2}}\langle\phi^{n-2}\chi^{m}\rangle+\frac{m(m-1)H^{3}}{8\pi^{2}}\langle\phi^{n}\chi^{m-2}\rangle\leavevmode\nobreak\
,$ (4.18)

where the second line comes from the drift and the third line is due to the
diffusion. This is an infinite set of differential equations for all the
correlators in the theory. For a free theory, the equations are decoupled and
only the diffusion terms remain on the right-hand side. Then it is easy to get
$\langle\phi^{2}\rangle=\langle\chi^{2}\rangle=H^{3}t/(2\pi)^{2}$, which is
just the Gaussian variance we computed in (4.4). However, for interesting
theories, this set of equations are not solvable in general. One needs to know
all of the equations even for computing lower-point correlation functions like
$\langle\phi^{2}\rangle$. Next, let’s try to make progress by treating the
drift terms perturbatively, and explicitly show the results for the three
types of interactions.

  * •

$V={\lambda\phi^{4}}/{4!}$ and $f(\chi)=0$: This simply corresponds to the
$\Phi^{4}$ interaction in the field-theoretic computation. For the single
scalar, the equations for non-vanishing correlation functions of $\phi$ are
given by

$\displaystyle\frac{d}{dt}\langle\phi^{2}\rangle$
$\displaystyle=-\frac{1\lambda}{9H}\langle\phi^{4}\rangle+\frac{H^{3}}{4\pi^{2}}$
$\displaystyle\frac{d}{dt}\langle\phi^{4}\rangle$
$\displaystyle=-\frac{2\lambda}{9H}\langle\phi^{6}\rangle+\frac{3H^{3}}{4\pi^{2}}\langle\phi^{2}\rangle$
$\displaystyle\qquad\qquad\vdots$
$\displaystyle\frac{d}{dt}\langle\phi^{n}\rangle$
$\displaystyle=-\frac{n\lambda}{18H}\langle\phi^{n+2}\rangle+\frac{n(n-1)H^{3}}{8\pi^{2}}\langle\phi^{n-2}\rangle\leavevmode\nobreak\
.$ (4.19)

Now we solve for the lower-point correlation functions order by order in
$\lambda$. The key point to note is that if we set $\lambda=0$, then the
solution for the correlation functions is approximately given by
$\langle\phi^{n}\rangle\sim H^{n}(\log a)^{n/2}$. Turning on $\lambda$
introduces corrections of $\lambda(\log a)^{n/2+1}$ to each $n$-point
correlator at the leading order. Then iteratively, the lower-point functions
get corrections from higher-point ones. For instance, truncating at
$\lambda^{3}$ is equivalent to setting $\lambda=0$ in the equation for
$\phi^{6}$. Doing so, we find that the two-point function becomes:

$\displaystyle\langle\phi^{2}\rangle=\frac{H^{2}}{4\pi^{2}}\log
a-\frac{\lambda H^{2}}{144\pi^{4}}(\log
a)^{3}+\frac{\lambda^{2}H^{2}}{2880\pi^{6}}(\log
a)^{3}+\mathcal{O}(\lambda^{3}(\log a)^{7}a)\leavevmode\nobreak\ .$ (4.20)

One can show that for corrections to the variance the expansion parameter is
$\lambda\log^{2}a$, and the $m$th order correction is proportional to
$\lambda^{m}(\log a)^{2m+1}$. This exactly agrees with what we have found in
field-theoretic computation for leading logarithms from classical loops.
Explicitly, we can check the result at one-loop level as it was obtained in
Eq.(4.8).

This calculation confirms that the contributions from classical loops are
incorporated within the stochastic formalism. One can further check that the
subleading terms from quantum loops, e.g. $\psi_{2}^{\mathrm{1-loop}}$, can be
associated with the corrections to the Fokker-Planck equation identified in
Ref. [12]. It is easy to see that these terms are subdominant in perturbative
expansions, as at the same order in $\lambda$ the quantum loops always
contribute fewer powers of logarithmic functions. We explicitly check this in
Appendix D where we perturbatively solve the noise term in the Fokker-Planck
equation. We get a contribution to the two point function that scales as
$\lambda(\log a)^{2}$ in accordance with the result obtained using the
perturbative wavefunction.

  * •

$V={g\phi^{3}}/{3!}$ and $f(\chi)=0$: This example is supposed to be related
to the $\Phi^{3}$ interaction. However, the potential is unbounded from below,
and the stochastic computation cannot be consistently applied. Thus we do not
make the explicit comparison with the field-theoretic computation from
$\Phi^{3}$ interaction here.

  * •

$f(\chi)=\alpha\chi^{2}/2$ and $V=0$. This corresponds to the two-field
example with the derivative interaction $\dot{\Phi}\Sigma^{2}$. The
differential equations for correlators are given by

$\displaystyle\frac{d}{dt}\langle\phi^{2}\rangle$
$\displaystyle=-\alpha\langle\phi\chi^{2}\rangle+\frac{H^{3}}{4\pi^{2}}$
$\displaystyle\frac{d}{dt}\langle\chi^{2}\rangle$
$\displaystyle=\frac{H^{3}}{4\pi^{2}}$
$\displaystyle\frac{d}{dt}\langle\phi\chi^{2}\rangle$
$\displaystyle=-\frac{\alpha}{2}\langle\chi^{4}\rangle+\frac{H^{3}}{4\pi^{2}}\langle\phi\rangle$
$\displaystyle\frac{d}{dt}\langle\chi^{4}\rangle$
$\displaystyle=\frac{3H^{3}}{2\pi^{2}}\langle\chi^{2}\rangle$
$\displaystyle\frac{d}{dt}\langle\phi\rangle$
$\displaystyle=-\frac{\alpha}{2}\langle\chi^{2}\rangle\leavevmode\nobreak\ .$

Solving these equations, we find the first-order correction to the two-point
function

$\displaystyle\langle\phi^{2}\rangle=\frac{H^{2}}{4\pi^{2}}\log
a+\frac{\alpha^{2}H^{2}}{128\pi^{4}}(\log a)^{4}\leavevmode\nobreak\ ,$ (4.21)

which again matches the result from classical loops obtained using the
wavefunction method as in Eq. (4.14). This case is actually simpler than the
$\phi^{4}$ and $\phi^{3}$ interactions. One way to see it is to notice that
there is a conserved current for interactions with one time derivative

$\displaystyle
j^{0}=\dot{\phi}-\frac{1}{2}f(\chi)=\frac{\pi_{\phi}}{a^{3}}\leavevmode\nobreak\
,$ (4.22)

which is a generalisation of the usual momentum conservation in the single
field case. So in the long wavelength limit $\dot{\phi}$ is no longer
conserved, but the more general combination given by $j_{0}$ is required to be
constant. Thus intuitively the massless scalar $\chi$ on the superhorizon
scales sources the growth of $\phi$, with $\phi\sim\alpha\chi^{2}\log a$, as
is normally seen in multi-field inflation models.

In the above solution, there seems to be no corrections at higher order in
$\alpha$. However, we should notice that when constructing the Hamiltonian due
to an interaction term like $\dot{\phi}f(\chi)$, additional terms involving
self-interactions of $\chi$ will inevitably appear. This is a consequence of
$\chi$ depending on momentum, as given by
$a^{-3}\pi_{\phi}=\dot{\phi}-\frac{1}{2}f(\chi)$. Consequently, the
interacting Hamiltonian will contain a term proportional to $f(\chi)^{2}/4$,
which can be conceptualized as an effective potential term, denoted as $V_{\rm
eff}(\phi)=f(\chi)^{2}/4$. The influence of this term is subdominant, at most
manifesting at the next-to-leading order of the couplings represented by
$f(\chi)$. Including this term perturbs the Fokker-Planck equation in a manner
akin to the situation when a potential like $V(\phi)=\lambda\phi^{4}$ is
considered. In the aforementioned example, the presence of the self-
interactions of $\chi$ alters the equation for $\langle\chi^{2}\rangle$,
consequently introducing corrections that begin at order $\alpha^{4}$ to the
two-point function $\langle\phi^{2}\rangle$. In this way we could derive the
higher order corrections systematically. It will be possible to show that the
result will scale as $(\alpha^{2}\log^{3}a)^{m}$ for the $m$th order, which is
in agreement with a similar estimate for the wavefunction computation of
classical loops from Eq.(3.24).

Now we are in the position to establish the connection between the
wavefunction approach and the stochastic formalism in perturbation theory.
From the above examples, we have seen that the correlation functions computed
through perturbation theory are the same as those computed by solving the
Fokker-Planck equation perturbatively. While we explicitly checked the matches
of results for the variance, the same analysis can be extended to higher-point
functions.

In general, at each order in perturbation theory, the stochastic computation
always leads to the same power of $\log a$ as that found from classical loops
in the wavefunction method. Meanwhile, the quantum loop contributions are
suppressed because they have less powers of logarithmic functions. This
observation provides a simple answer about what the stochastic formalism is
resumming: classical loops from tree-level wavefunction coefficients. It
confirms our intuitive understanding that IR divergences in dS are governed by
semi-classical physics on superhorizon scales.

Interestingly, it also explains why the Fokker-Planck equation may not be a
good description for IR-finite theories, such as the ones with higher-
derivative interaction: there the quantum loops are not always suppressed and
can be equally important as the classical loop contributions, while they are
beyond the scope of a stochastic description.

In the end, we would like to comment on the differences with Ref. [13] where
similar results about leading IR divergences at any loop order were derived.
There the authors applied the retarded in-in formalism to identify the leading
logarithms in perturbaton theory, and in the end “the resummation of time”
reproduces results from the Fokker-Planck equation. Here with the wavefunction
method, we further point out that the these leading logarithms have an origin
as classical loops. This new understanding helps us to simplify the
perturbative computation for IR divergences at the loop level. Meanwhile, it
indicates that the leading IR effects should be captured by the saddle-point
approximation of the wavefunction (2.6), as all the tree-level information has
been included there. With this insight, we shall go beyond the perturbation
theory in the next section and directly study the implications of the semi-
classical wavefunction for IR divergences in dS.

##  5 Beyond Perturbation Theory

In this section, we go beyond the perturbative regime of IR divergences in dS,
and consider how the stochastic formalism can be derived from the semi-
classical wavefunction. We will show that the Fokker-Planck equation follows
as a consequence of the Schrödinger equation and Polchinski’s exact
renormalization group equation from a boundary perspective.

As we noticed in the previous sections, when we have IR divergences in de
Sitter, the semi-classical wavefunction always provides the dominant
contributions to correlators in perturbation theory, which match the
perturbative computation in the stochastic formalism. This interesting
observation suggests that the stochastic formalism, which is also expected to
be valid in the non-perturbative regime, actually resums the classical loops
from all the tree-level wavefunction coefficients. In this section we shall
show that indeed the Fokker-Planck equation can be derived from the saddle-
point approximation of the wavefunction.

Our focus lies in the coarse-grained theory at the late-time boundary, where
we can effectively “integrate out” the short-wavelength modes. This process
results in an effective description for the long-wavelength modes. Explicitly,
we regularize the theory by requiring that only momentum modes up to the UV
cutoff scale $\Lambda$ are allowed. Then in analogy with (2.3), the coarse-
grained version of the wavefunction can be expressed as a functional of the
boundary field

$\displaystyle\Psi_{\Lambda}[\phi,t]=$
$\displaystyle\exp\left[\frac{1}{2}\int_{{\bf
k}}\psi_{2}(k)\Omega_{\Lambda}^{-1}(k)\leavevmode\nobreak\ \phi_{{\bf
k}}\phi_{-{\bf k}}+\sum_{n=3}^{\infty}\frac{1}{n!}\int_{{\bf k}_{1},...,{\bf
k}_{n}}\,\psi_{n}^{\Lambda}({\bf k}_{1},...,{\bf k}_{n},t)\leavevmode\nobreak\
\phi_{{\bf k}_{1}}\cdot\cdot\cdot\phi_{{\bf k}_{n}}\,\right]\,.$ (5.1)

Here in the free theory part $\psi_{2}$, we have used the window function
$\Omega_{\Lambda}(k)$ introduced below (4.1) for the blocking. Now we see this
window function fixes the $\Lambda$-dependence of the quadratic term in
$\Psi_{\Lambda}$, while the concrete forms of $\psi_{n\geq 3}^{\Lambda}$
remain unclear, especially when we go beyond perturbation theory. This is an
important question that will be analyzed in the following derivation using the
RG flow techniques. In the end we recall that this cutoff scale depends on the
time of the boundary slice $\Lambda(t)=\epsilon a(t)H$ with $\epsilon\ll 1$.

Next, we consider the probability distribution of field fluctuations
$\phi({\bf x})$ at the late-time boundary $t$. By using the coarse-grained
wavefunction, we simply have

$P_{\Lambda}[\phi,t]=|\Psi_{\Lambda}[\phi,t]|^{2}=e^{W_{0}[\phi]+W_{I}[\phi]}\leavevmode\nobreak\
,$ (5.2)

with the free and interacting parts given by

$\displaystyle W_{0}$ $\displaystyle=$ $\displaystyle\int_{{\bf k}}{\rm
Re}\leavevmode\nobreak\
\psi_{2}(k)\Omega_{\Lambda}^{-1}(k)\leavevmode\nobreak\ \phi_{{\bf
k}}\phi_{-{\bf k}}\leavevmode\nobreak\ ,$ $\displaystyle W_{I}$
$\displaystyle=$ $\displaystyle\sum_{n=3}^{\infty}\frac{1}{n!}\int_{{\bf
k}_{1},...,{\bf k}_{n}}\,2{\rm Re}\leavevmode\nobreak\ \psi_{n}^{\Lambda}({\bf
k}_{1},...,{\bf k}_{n},t)\leavevmode\nobreak\ \phi_{{\bf
k}_{1}}\cdot\cdot\cdot\phi_{{\bf k}_{n}}\,.$ (5.3)

To derive the Fokker-Planck equation, we notice that the time dependence of
$P_{\Lambda}$ comes from two different terms. The first one is from the
wavefunction itself: as we have seen in the perturbative computation with IR
divergences, the wavefunction coefficients in general contain secular terms
with logarithmic growth. The second one comes from the time dependence of the
cutoff $\Lambda(t)$. Thus we find

$\frac{d}{dt}P_{\Lambda}[\phi,t]=\frac{\partial}{\partial
t}P_{\Lambda}[\phi,t]+\dot{\Lambda}\frac{\partial}{\partial\Lambda}P_{\Lambda}[\phi,t]\leavevmode\nobreak\
.$ (5.4)

In the rest of the section, we shall look into these two contributions and
focus on their physical origins. Before going into the details of the
derivation, the upshot is that the first term gives the classical drift from
the Schrödinger equation, while the second term comes as a RG flow on the
boundary which in the end leads to the diffusion. Some of our derivation share
similarities with the one presented in Ref. [12], especially for the
derivation of the drift term. The novelty in our approach is to notice that in
the semi-classical regime the diffusion term comes as a consequence of
Polchinski’s equation of RG flow on the boundary. We shall comment on the
differences with Ref. [12] in more detail at the end of this section.

###  5.1 Diffusion as the RG Flow on the Boundary

Let’s first briefly review Polchinski’s derivation of the exact
renormalization equation for a general quantum field theory in $d$-dimensional
space. The basic idea of the exact RG flow starts from the requirement that
physics should be independent of the artificial cutoff scale that we imposed
in the theory, thus the partition function remains invariant with respect to
the change of $\Lambda$

$\Lambda\frac{d}{d\Lambda}Z_{\Lambda}=0\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
with}\leavevmode\nobreak\ \leavevmode\nobreak\
Z_{\Lambda}\equiv\int\mathcal{D}\varphi\leavevmode\nobreak\ e^{-S_{\rm
eff}^{\Lambda}}\leavevmode\nobreak\ ,$ (5.5)

where $S_{\rm eff}^{\Lambda}$ is the effective action below the scale
$\Lambda$ with UV degrees of freedom being integrated out. Then as we change
the UV cutoff, the overall partition function should be unaffected, which
basically means that, the low-energy effective action would vary
correspondingly to account for the change in the short-wavelength modes
($k>\Lambda$) that are being integrated out. Specifically, Polchinski’s idea
is to cut off the theory with a smooth window function $\Omega_{\Lambda}$ as
the one introduced in (5.1). Then the effective action can be written as the
free theory part and the interacting part $S_{\rm
eff}^{\Lambda}=S_{0}^{\Lambda}+S_{\rm int}^{\Lambda}$, with

$S_{0}^{\Lambda}=\frac{1}{2}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{\Omega_{\Lambda}}\varphi_{\bf
k}G_{k}^{-1}\varphi_{-\bf k}\leavevmode\nobreak\ .$ (5.6)

Here $G_{k}$ is the propagator in momentum space, and in the above expression
we see that it has been regularized by the smooth window function
$\Omega_{\Lambda}(k)$. Then the condition (5.5) in the end leads to a non-
perturbative functional differential equation for the $\Lambda$-dependence of
$S_{\rm int}^{\Lambda}$

$\Lambda\frac{d}{d\Lambda}e^{-S_{\rm
int}^{\Lambda}}=-\frac{1}{2}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}{G_{k}}\frac{\delta^{2}}{\delta\varphi_{\bf
k}\delta\varphi_{-\bf k}}e^{-S_{\rm int}}\leavevmode\nobreak\ .$ (5.7)

By substituting (5.7) into (5.5) and performing integration by parts twice,
one can check that the $\Lambda$-dependence of $S_{\rm int}^{\Lambda}$ cancels
out the $\Lambda$-dependence of $S_{0}^{\Lambda}$, and thus the partition
function remains independent of $\Lambda$. Equation (5.7) is known as
Polchinski’s exact renormalization group [81], which provides a general and
non-perturbative description for the running of couplings in the low-energy
effective theory. See [82, 83, 84, 85, 86] for recent reviews and discussions.
It is worth mentioning that, Polchinski’s equation (5.7) is not the only
option to satisfy the condition in (5.5). As integrations by parts are
included, we are left with the freedom to add boundary terms in (5.7) without
affecting the $\Lambda$-independence of $Z_{\Lambda}$. These boundary terms
correspond to different choices of RG schemes.

Similarly, for theories with IR divergences in dS, an effective description
for long wavelength modes is given by the coarse-grained wavefunction (5.1).
For now let us forget about the time evolution in the bulk, and consider this
3-dimensional Euclidean field theory on the boundary. As the object of
interest here is the probability distribution (5.2), we start with the
requirement that the conservation of probability
$\int\mathcal{D}\phi\leavevmode\nobreak\ P_{\Lambda}=1$ should be unaffected
by changing the cutoff scale

$\frac{d}{d\ln\Lambda}\int\mathcal{D}\phi\leavevmode\nobreak\
P_{\Lambda}[\phi,t]=\int\mathcal{D}\phi\leavevmode\nobreak\
\left[\frac{dW_{0}}{d\ln\Lambda}e^{W_{I}}+\frac{de^{W_{I}}}{d\ln\Lambda}\right]e^{W_{0}}=0\leavevmode\nobreak\
.$ (5.8)

This means that the $\Lambda$-dependence of the free theory part should be
cancelled out by the $\Lambda$-dependence of the interacting part. For
$W_{0}$, it depends on the cutoff scale through the window function

$\frac{dW_{0}}{d\ln\Lambda}=\int_{{\bf k}}{\rm Re}\leavevmode\nobreak\
\psi_{2}(k)\leavevmode\nobreak\ \phi_{{\bf k}}\phi_{-{\bf
k}}\frac{d}{d\ln\Lambda}\Omega_{\Lambda}^{-1}(k)\leavevmode\nobreak\ .$ (5.9)

For the $\Lambda$-dependence of $W_{I}$, there are also various choices that
can ensure the probability conservation with varying cutoff scales. In analogy
with (5.7), we propose the following version of Polchinski’s equation

$e^{W_{0}}\frac{de^{W_{I}}}{d\ln\Lambda}=\frac{1}{4}\int_{\bf
k}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}\frac{1}{{\rm
Re}\psi_{2}}\left[\left(\frac{\delta^{2}}{\delta\phi_{\bf k}\delta\phi_{-\bf
k}}e^{W_{I}}\right)e^{W_{0}}-2\frac{\delta}{\delta\phi_{\bf
k}}\left(e^{W_{0}}\frac{\delta e^{W_{I}}}{\delta\phi_{-\bf
k}}\right)\right]\leavevmode\nobreak\ ,$ (5.10)

which has an extra boundary term. This is supposed to be the exact RG equation
on the late-time boundary for the semi-classical wavefunction. Let us justify
it step by step. First of all, it is straightforward to check that with this
ansatz indeed (5.8) is satisfied. By substituting (5.10) into (5.8), we find

$\displaystyle\frac{d}{d\ln\Lambda}\int\mathcal{D}\phi\leavevmode\nobreak\
P_{\Lambda}[\phi,t]$ $\displaystyle=$ $\displaystyle\frac{1}{4}\int_{\bf
k}\frac{1}{{\rm Re}\psi_{2}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}\int
D\phi\left[-\frac{\delta^{2}e^{W_{0}}}{\delta\phi_{\bf k}\delta\phi_{-\bf
k}}e^{W_{I}}\right.$ (5.11) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\left.+\left(\frac{\delta^{2}}{\delta\phi_{\bf k}\delta\phi_{-\bf
k}}e^{W_{I}}\right)e^{W_{0}}-2\frac{\delta}{\delta\phi_{\bf
k}}\left(e^{W_{0}}\frac{\delta e^{W_{I}}}{\delta\phi_{-\bf
k}}\right)\right]\leavevmode\nobreak\ ,$

where the first term comes from rewriting (5.9). The last term is a total
functional derivative and thus can be neglected. Then doing integration by
parts twice for the second term in the bracket, we move the
$\frac{\delta}{\delta\phi}$ derivatives on $e^{W_{I}}$ onto $e^{W_{0}}$.
Neglecting the boundary terms, we see that the remaining piece just cancels
out the first term from the $\Lambda$-dependence of $W_{0}$.

Next, we need to justify the appearance of the last term in (5.10). In
general, we have the freedom to add arbitrary boundary terms in Polchinski’s
equation, which correspond to different RG schemes. The choice in (5.10)
corresponds to the semi-classical scheme that will be spelled out below. As we
shall see soon after, this particular form of the boundary term in (5.10)
becomes important for deriving the diffusion term in the Fokker-Planck
equation.

#### The semi-classical scheme

So far we have kept our analysis fully non-perturbative, while the RG flow
equation (5.10) is also expected to be valid in the perturbative regime.
Equivalently, we can write it as the following flow equation satisfied by the
interacting part of the wavefunction

$\frac{dW_{I}}{d\ln\Lambda}=-\frac{1}{4}\int_{\bf k}\frac{1}{{\rm
Re}\psi_{2}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}\left[\frac{\delta^{2}{W_{I}}}{\delta\phi_{\bf
k}\delta\phi_{-\bf k}}+\frac{\delta{W_{I}}}{\delta\phi_{\bf
k}}\frac{\delta{W_{I}}}{\delta\phi_{-\bf
k}}+2\frac{\delta{W_{0}}}{\delta\phi_{\bf
k}}\frac{\delta{W_{I}}}{\delta\phi_{-\bf k}}\right]\leavevmode\nobreak\ ,$
(5.12)

which tells us the running of wavefunction coefficients with the cutoff scale
in interacting theories.101010It is an analogue to the running of couplings in
the standard quantum field theory. Here we choose the couplings, like $g$ and
$\lambda$, to be constant, and the $\Lambda$-dependence comes from the
logarithmic terms in the coarse-grained wavefunction coefficients. Therefore,
one way to think about this running is to consider the effective couplings
that absorb the logarithmic $\Lambda$-dependence in $\psi_{n}$. Now we
explicitly show that the flow equation has to be the one above in order to
match the perturbation theory computation with the semi-classical
wavefunction.

For concreteness, we take the $g\Phi^{3}$ interaction as an example. At the
$g$ and $g^{2}$ orders, we have the following components in the semi-classical
wavefunction

$W_{I}^{(1)}=\frac{1}{3}\int_{\bf k_{1},k_{2},k_{3}}{\rm
Re}\psi_{3}^{\Lambda}\leavevmode\nobreak\ \phi_{\bf k_{1}}\phi_{\bf
k_{2}}\phi_{\bf k_{3}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ W_{I}^{(2)}=\frac{1}{12}\int_{\bf
k_{1},k_{2},k_{3}}{\rm Re}\psi_{4}^{\Lambda}\leavevmode\nobreak\ \phi_{\bf
k_{1}}\phi_{\bf k_{2}}\phi_{\bf k_{3}}\phi_{\bf k_{4}}\leavevmode\nobreak\ ,$
(5.13)

which correspond to the three-point contact and four-point single-exchange