Stochastic Inflation at NNLO

Timothy Cohen${}^{\hskip 0.8pt1}$, Daniel Green${}^{\hskip 0.8pt2}$, Akhil
Premkumar${}^{\hskip 0.8pt2}$, and Alexander Ridgway${}^{\hskip 0.8pt2}$

1Institute for Fundamental Science, University of Oregon, Eugene, OR 97403,
USA 2Department of Physics, University of California at San Diego, La Jolla,
CA 92093, USA

Abstract

Stochastic Inflation is an important framework for understanding the physics
of de Sitter space and the phenomenology of inflation. In the leading
approximation, this approach results in a Fokker-Planck equation that
calculates the probability distribution for a light scalar field as a function
of time. Despite its successes, the quantum field theoretic origins and the
range of validity for this equation have remained elusive, and establishing a
formalism to systematically incorporate higher order effects has been an area
of active study. In this paper, we calculate the next-to-next-to-leading order
(NNLO) corrections to Stochastic Inflation using Soft de Sitter Effective
Theory (SdSET). In this effective description, Stochastic Inflation manifests
as the renormalization group evolution of composite operators. The leading
impact of non-Gaussian quantum fluctuations appears at NNLO, which is
presented here for the first time; we derive the coefficient of this term from
a two-loop anomalous dimension calculation within SdSET. We solve the
resulting equation to determine the NNLO equilibrium distribution and the low-
lying relaxation eigenvalues. In the process, we must match the UV theory onto
SdSET at one-loop order, which provides a non-trivial confirmation that the
separation into Wilson-coefficient corrections and contributions to initial
conditions persists beyond tree level. Furthermore, these results illustrate
how the naive factorization of time and momentum integrals in SdSET no longer
holds in the presence of logarithmic divergences. It is these effects that
ultimately give rise to the renormalization group flow that yields Stochastic
Inflation.

###### Contents

  1. 1 Introduction
  2. 2 Stochastic Inflation
    1. 2.1 Leading Order
    2. 2.2 Beyond Leading Order
  3. 3 Soft de Sitter Effective Theory
    1. 3.1 In-In Correlators
    2. 3.2 Taking the Long Wavelength Limit
    3. 3.3 (Dynamical) Renormalization
    4. 3.4 Matching and Initial Conditions
    5. 3.5 Stochastic Inflation from SdSET
  4. 4 Matching $\lambda\phi^{4}$ Onto SdSET at One-Loop
    1. 4.1 Tree-level Matching and Field Redefinitions
    2. 4.2 One-loop Matching
    3. 4.3 Initial Conditions for Composite Operators
  5. 5 Composite Operator Mixing
    1. 5.1 One-loop Corrections
    2. 5.2 Two-loop Corrections
    3. 5.3 Stochastic Inflation at NNLO
  6. 6 Implications
    1. 6.1 Equilibrium Probability Distribution
    2. 6.2 Relaxation Eigenvalues
  7. 7 Conclusions
  8. A Matching the Six-point Function
  9. B Hard Cutoff Calculations
    1. B.1 Matching
    2. B.2 Composite Operator Mixing

##  1 Introduction

The study of quantum fields in de Sitter (dS) space provides insight into the
foundations of inflationary cosmology. In particular, the equal-time in-in
correlation functions of light scalar fields form the theoretical
underpinnings of the predictions for the observed density fluctuations sourced
during inflation [1, 2, 3]. These correlators encode a wealth of information
about the inflationary era that could be revealed by measurements of
primordial non-Gaussianity [4]. Yet, despite their importance, our
understanding of cosmological correlators beyond tree level is quite limited.
For light scalars, explicit loop calculations have revealed the presence of
infrared (IR) divergences and unbounded time-dependent “secular” growth [5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

Stochastic Inflation [21, 22, 23] is a framework for treating the IR dynamics
of a massless scalar field in a dS background, and has long been suspected to
provide the non-perturbative resolution to the IR issues associated with
massless fields in dS [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. The
idea is to reframe the problem in terms of the probability distribution for
the scalar field as a function of time, resulting in a Fokker-Planck equation
that depends on the scalar field potential. There are two contributions to the
evolution, resulting from quantum noise induced by fluctuations of the field
as it crosses the dS horizon and classical drift due to the potential. This
framework forms the conceptual basis for slow-roll eternal inflation [36, 37,
38] and can be used to describe the onset of eternal inflation quantitatively
[39]. Moreover, these results hint at the physical meaning of the dS entropy
[40, 41, 42], which remains a significant unsolved problem [43].

Notwithstanding the conceptual and technical appeal of Stochastic Inflation,
it is necessarily approximate. For example, we expect that there are non-
Gaussian contributions to the quantum noise that result from the UV
interaction, which are not modeled by Stochastic Inflation. Furthermore, the
Fokker-Planck formalism obscures the connection to cosmological correlators,
and it is not a prioi obvious how to incorporate higher-order corrections. It
would be ideal if we could understand how the success of Stochastic Inflation
relates to other results regarding the IR behavior of fields in dS, such as
the freeze-out of superhorizon metric fluctuations which has been shown to all
orders in perturbation theory [44, 45, 46], or the loop generated anomalous
scaling for the time-evolution of massive fields [47, 48]. One of our goals in
exploring the corrections to Stochastic Inflation is to understand how they
fit into the broader context of quantum field theory in dS.

A framework that accomplishes this ambitious goal is the Soft de Sitter
Effective Theory (SdSET) [33]. By following the standard Effective Field
Theory (EFT) playbook, this approach isolates the dynamics that persist in the
superhorizon limit, yielding more efficient calculations of loop corrections
to long wavelength cosmological correlators. Taking a real scalar field in dS
as the UV description, SdSET describes the IR limit of this model by relying
on two degrees of freedom that correspond to the growing and decaying modes
which are familiar from the solving the Klein-Gordon equation classically in a
dS background. This representation admits a power counting prescription that
systematically expands about the long wavelength limit in terms of a local
Lagrangian. Loop dependence on time and space is manifestly factorized
throughout the calculation, allowing an efficient isolation of the time
dependent IR divergent logs. Such logs lead to secular growth for both massive
and light fields, and appear in SdSET as contributions to the anomalous
dimensions of local operators. In the case of light fields, an infinite number
of operators become degenerate and Starobinsky’s model of Stochastic Inflation
is equivalent to the leading order (LO) dynamical renormalization group (RG)
that governs their mixing as a function of time.111The dynamical RG flow
described here should not be confused with the RG flow that appears in a
holographic dual via dS/CFT [43, 49, 50, 1, 51]. The key difference is that
our dynamical RG applies directly to the in-in correlators and not to the
wavefunction of the universe. This implies that corrections to Stochastic
Inflation can be computing by simply extending the RG analysis to higher
orders.

Taking the UV description to be massless $\lambda\phi^{4}$ theory, the
endpoint of this RG flow is a non-trivial fixed point where the field values
are $\phi\sim H\lambda^{-1/4}$. Corrections to this description around this
fixed point must account for this non-perturbative scaling with $\lambda$. In
this paper, we will calculate the evolution of operators to next-to-next-to
leading order (NNLO) in this power counting. At NLO, our results reproduce
previous calculations [30, 34]; as we will show, these contributions can be
attributed to field definitions within SdSET. In contrast, at NNLO we find a
universal correction in the form of a two-loop anomalous dimension that
introduces the first higher derivative correction to Stochastic Inflation. In
the process, we perform the full one-loop matching in SdSET, which further
elucidates the relationship between the EFT and the UV descriptions.

One novel feature of SdSET is that consistently matching a UV theory onto the
EFT requires specifying both Wilson coefficients and (time-independent)
initial conditions. Deriving the RG that yields Stochastic Inflation at NNLO
requires performing this matching explicitly at one-loop order. This provides
a highly non-trivial check of the SdSET formalism, and these results can be
utilized for a wide variety of correlator calculations. We will also use this
calculation as an opportunity to demonstrate the power of the symmetry
preserving “dynamical dimensional regularization” technique introduced in
[33].

This paper is organized as follows. We begin with a review of Stochastic
Inflation in Sec. 2, with an emphasis on its origins as a Markovian process,
which provides a framework with which we can organize corrections. Then Sec. 3
reviews the most salient aspects of the SdSET formalism. The new calculations
begin in Sec. 4, where we present the one-loop matching results that are
relevant for our applications here. These are then applied in Sec. 5, where we
compute the composite operator anomalous dimensions that feed into Stochastic
Inflation up to NNLO, and leads to the main result of this work in Eq. 5.33.
We then explore the implications of this formula in Sec. 6, and finally
conclude in Sec. 7. An appendix on the relevant, but somewhat technical, six-
point function matching is provided in App. A, and the hard cutoff version of
the main calculations are given in App. B.

Guide for the reader: We have attempted to make this paper accessible to a
wide ranging audience. For the casual reader, we suggest ($i$) reading Sec. 2
and Sec. 3.5, ($ii$) studying the matching diagrams in Sec. 4, and ($iii$)
studying the composite operator mixing diagrams in Sec. 5. This brings the
reader to ($iv$) the nearly final result in Eq. 5.25, where the explicit
corrections are given in Eq. 5.26. Finally, ($v$) Sec. 5.3 provides the
derivation that results in the simplified form of the NNLO formula for
Stochastic Inflation given in Eq. 5.33. One might also be interested in the
NNLO equilibrium probability distribution and relaxation eigenvalues presented
in Sec. 6, which brings the reader to the conclusions in Sec. 7.

##  2 Stochastic Inflation

In a theory of a massless scalar field $\phi$ in a dS background with Hubble
constant $H$, the field’s value will fluctuate by $O(H)$ as each momentum mode
crosses the dS horizon. An equivalent point of view is that these stochastic
fluctuations are the result of the non-zero temperature within dS. This effect
has a natural interpretation as a random walk, an idea that was made precise
by Starobinsky [21] and led the formalism known as Stochastic Inflation. In
this section, we discuss this approach, and emphasize the structure of higher
order corrections.

For concreteness, we will assume the canonical example of $\lambda\phi^{4}$
theory in a dS background, whose UV description in terms of a scalar field
$\phi$ is

$S_{\rm UV}=\int{\rm d}^{4}x\,\sqrt{-g}\,\bigg{[}-\frac{1}{2}\hskip
0.8ptg_{\mu\nu}\hskip
0.8pt\partial^{\mu}\phi\,\partial^{\nu}\phi+\frac{1}{2}\hskip 0.8ptm^{2}\hskip
0.8pt\phi^{2}+\frac{1}{4!}\hskip 0.8pt\lambda\hskip 0.8pt\phi^{4}\hskip
0.8pt\bigg{]}\,,$ (2.1)

where $g_{\mu\nu}$ is the dS metric and $g\equiv\det g_{\mu\nu}$ as usual. The
essential formalism developed here holds for general models. However, we will
not be able to derive corrections to Stochastic Inflation generically, and so
we will work with this simple and well studied example when we calculate
explicit corrections.

In the process of discussing the general structure of corrections to
Stochastic Inflation, we will arrive at a natural interpretation for higher-
order corrections in terms of the transition amplitudes for the field $\phi$.
Unfortunately, how to determine the corrections directly is not transparent in
this description. In Sec. 3.5 we will show how the corrections discussed in
this Section arise from operator mixing, and the remainder of the paper is
devoted to deriving these corrections and their implications using SdSET.

Before moving on, we emphasize that the formalism we develop in this section
will rely on the assumption that the late time evolution of $\phi$ can be
modeled as a Markovian system (as described in Sec. 2.2 below). This will be
justified by the concrete calculation of the dynamical renormalization group
using the SdSET that is developed later in this paper. It ultimately is due to
the fact that the dynamics of the SdSET degrees of freedom are governed by a
first order equation, which is a consequence of the EFT power counting.

###  2.1 Leading Order

The framework of Stochastic Inflation results in a probability distribution
$P(\phi,t)$ for the field $\phi$ at a time $t$. To leading order, $P(\phi,t)$
obeys a Fokker-Planck equation

$\frac{\partial}{\partial
t}P(\phi,t)=\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}}{\partial\phi^{2}}P(\phi,t)+\frac{1}{3H}\frac{\partial}{\partial\phi}\big{[}V^{\prime}(\phi)P(\phi,t)\big{]}\,,$
(2.2)

where the first term captures the stochastic noise from the inherent quantum
variance of $\phi$, while the second term is due to the classical drift
induced by the potential where $V^{\prime}(\phi)\equiv\partial
V/\partial\phi$. One interesting application of this equation is to solve for
the fixed point that the scalar field would reach if it lived in an eternal dS
background. To find the fixed point, we enforce that $\partial P_{\rm
eq}/\partial t=0$ for the equilibrium solution $P_{\rm eq}$, which implies

$\frac{\partial^{2}}{\partial\phi^{2}}P_{\rm
eq}(\phi)=-\frac{8\pi^{2}}{3H^{4}}\frac{\partial}{\partial\phi}\big{[}V^{\prime}(\phi)P_{\rm
eq}(\phi)\big{]}\,.$ (2.3)

Integrating both sides of this equation twice leads to the solution

$P_{\rm eq}(\phi)=Ce^{-8\pi^{2}V(\phi)/3H^{4}}\,.$ (2.4)

We can use this leading order solution to organize corrections to Stochastic
Inflation as a perturbative series in the UV coupling.

###  2.2 Beyond Leading Order

Having reviewed the leading order formalism and its consequence for pure dS,
we now turn to exploring the form we can expect corrections to take. In order
to generalize this Fokker-Planck equation, we return to its origins. The
underlying assumption is that the system is Markovian, in that the time slice
of interest is entirely determined by the information contained in the
previous step. In other words, a Markovian system has no “memory.” Since this
assumption holds for the spectrum of scalar field fluctuations at horizon
crossing, the resulting formalism will tell us what kinds of corrections to
Stochastic Inflation we can expect.

The Markovian assumption leads directly to the Chapman-Kolmogorov equations,
which describe a probability distribution $P(\phi,t+{\rm d}t)$ that is fully
determined by $P(\phi,t)$:

$\frac{\partial}{\partial t}P(\phi,t)=\int{\rm
d}\phi^{\prime}\,\Big{[}P(\phi^{\prime},t)\hskip
0.8ptW(\phi|\phi^{\prime})-P(\phi,t)\hskip
0.8ptW(\phi^{\prime}|\phi)\Big{]}\,,$ (2.5)

where $W(\phi|\phi^{\prime})$ is a “transition rate” in that it sets the rate
for transitioning to $\phi$ from another value $\phi^{\prime}$ in a
differential amount of time. This equation simply expresses that the
probability distribution for $\phi$ at $t+{\rm d}t$ is fully determined by the
weighted sum of the possible transitions that yield $\phi$ minus the sum of
all the weighted transitions for $\phi$ to change value.

$\phi$$W(\phi^{\prime\prime}|\phi)$$W(\phi^{\prime}|\phi)$$W(\phi|\phi^{\prime\prime})$$W(\phi|\phi^{\prime})$$\phi$$\widetilde{W}(\Delta\phi|\phi-\Delta\phi)$$\widetilde{W}(\Delta\phi|\phi)$$\Delta\phi$$\Delta\phi$
Figure 1: Visualization of the Kramers-Moyal expansion. The top panel shows
the probability of “hopping” from $\phi$ to $\phi^{\prime}$,
$W(\phi^{\prime}|\phi)$ or from $\phi^{\prime}$ to $\phi$,
$W(\phi|\phi^{\prime})$, or equivalently any other point such as
$\phi^{\prime\prime}$. On the bottom, we see the process in terms of the
probability $\widetilde{W}(\Delta\phi|\phi)$ to hop from a specific starting
point $\phi$ by a distance $\Delta\phi$.

Next, we will reorganize Eq. 2.5 using what is known as the Kramers-Moyal
expansion, visualized in Fig. 1. This is effectively the assumption that the
transitions are dominated by “local” jumps [52]. The first step is to make a
substitution of $\Delta\phi=\phi-\phi^{\prime}$ in the first term and
$\Delta\phi=\phi^{\prime}-\phi$ in the second term. In other words, when
$\phi$ is in the final state, then $\phi^{\prime}=\phi-\Delta\phi$ and when
$\phi^{\prime}$ is in the final state $\phi^{\prime}=\phi+\Delta\phi$. This
yields

$\frac{\partial}{\partial t}P(\phi,t)=\int{\rm
d}\Delta\phi\Big{[}P(\phi-\Delta\phi,t)\hskip
0.8ptW(\phi|\phi-\Delta\phi)-P(\phi,t)\hskip
0.8ptW(\phi+\Delta\phi|\Delta\phi)\Big{]}\,,$ (2.6)

where we have included two compensating relative minus signs, one from the
different changes of variables for the two terms, and another due to needing
to flip the limits of integration after switched the integration variable
$\Delta\phi\rightarrow-\Delta\phi$ in the second term. Since we are assuming
the main support of $W$ comes from local jumps, we want to Taylor expand the
first term for fixed $\phi$. To make performing this expansion transparent, it
is then useful to redefine $W$ using

$\displaystyle\widetilde{W}(y,x)\equiv W(x+y|x)\,,$ (2.7)

so that Eq. 2.6 becomes

$\displaystyle\frac{\partial}{\partial t}P(\phi,t)=\int{\rm
d}\Delta\phi\Big{[}P(\phi-\Delta\phi,t)\hskip
0.8pt\widetilde{W}(\Delta\phi,\phi-\Delta\phi)-P(\phi,t)\hskip
0.8pt\widetilde{W}(\Delta\phi,\phi)\Big{]}\,.$ (2.8)

Then Taylor expanding the first term about a fixed value of $\phi$ yields

$\displaystyle\int{\rm d}\Delta\phi\,P(\phi-\Delta\phi,t)\hskip
0.8pt\widetilde{W}(\Delta\phi,\phi-\Delta\phi)$ $\displaystyle=\int{\rm
d}\Delta\phi\sum_{n=0}^{\infty}\frac{1}{n!}\bigg{(}\\!\\!-\\!\Delta\phi\frac{\partial}{\partial\phi}\bigg{)}^{n}P(\phi,t)\hskip
0.8pt\widetilde{W}(\Delta\phi,\phi)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\partial^{n}}{\partial\phi^{n}}\int{\rm
d}\Delta\phi\,\big{(}\\!-\\!\Delta\phi\big{)}^{n}\,P(\phi,t)\widetilde{W}(\Delta\phi,\phi)\,,$
(2.9)

where in the last line, we used the fact that $\Delta\phi$ is independent of
$\phi$ to pull the derivatives outside of the integral. Plugging this
expansion into Eq. 2.8, we see that the $n=0$ term cancels so that

$\frac{\partial}{\partial
t}P(\phi,t)=\sum_{n=1}^{\infty}\frac{1}{n!}\frac{\partial^{n}}{\partial\phi^{n}}\,\Omega_{n}(\phi)\hskip
0.8ptP(\phi,t)\,,$ (2.10)

where

$\Omega_{n}(\phi)\equiv\int{\rm
d}\Delta\phi\,\big{(}\\!-\\!\Delta\phi\big{)}^{n}\,W(\phi+\Delta\phi|\Delta\phi)\,,$
(2.11)

and we have used Eq. 2.7 to write this expression in terms of the original
transition rate $W$. Note that all terms in this expansion are total
derivatives, as required for the conservation of the total probability. We
also see that $\Omega_{n}(\phi)$ encodes the $\phi$-dependence of the
$n^{\text{th}}$ moment of the distribution. For the theories of interest here,
$\Omega_{n}(\phi)$ will admit a polynomial expansion in $\phi$:

$\displaystyle\Omega_{n}(\phi)=\sum_{m=0}^{\infty}\frac{1}{m!}\Omega_{n}^{(m)}\phi^{m}\,,$
(2.12)

for some coefficients $\Omega_{n}^{(m)}$.

Thus far, all of this discussion was very general. If we specialize to the
case of leading order Stochastic Inflation, we can compare this expanded
result with Eq. 2.2 to identify that the $n=1$ “drift” term is proportional to
the derivative of the potential,

$\displaystyle V=\sum_{\ell}\frac{1}{\ell!}c_{\ell}\phi^{\ell}\,.$ (2.13)

so that matching with Eq. 2.2 implies

$\displaystyle\Omega_{1}^{(m)}=\frac{1}{3H}c_{m+1}\,.$ (2.14)

Hence, for $n=1$, $m$ tracks the polynomial interactions that could appear in
a generic potential. Moving to the $n=2$ “noise” term, we again can compare to
Eq. 2.2 to find

$\displaystyle\Omega_{2}^{(0)}=\frac{H^{3}}{4\pi^{2}}\,.$ (2.15)

In this case, the $m>0$ terms correspond to higher order corrections.

To summarize, if we assume that the UV theory has only a $\lambda\phi^{4}$
interaction, we conclude that the generalized evolution equation that
describes Stochastic Inflation takes the form

$\displaystyle\frac{\partial}{\partial
t}P(\phi,t)=\sum_{n=2}^{\infty}\frac{1}{n!}\frac{\partial^{n}}{\partial\phi^{n}}\bigg{[}\sum_{m=0}^{\infty}\frac{1}{m!}\Omega_{n}^{(m)}\phi^{m}\,P(\phi,t)\bigg{]}+\frac{1}{3H}\frac{\partial}{\partial\phi}\bigg{[}V^{\prime}(\phi)P(\phi,t)\bigg{]}\,,$
(2.16)

where $V^{\prime}$ is the $\phi$ derivative of the potential, which includes
the matching corrections required to obtain the accuracy of interest.

To develop some intuition for what the $\Omega_{n}$ corrections are capturing,
we can interpret $W(\phi+\Delta\phi|\Delta\phi)$ as the probability
distribution of transitions of size $\Delta\phi$. Then $\Omega_{n}(\phi)$ is
simply the $n^{\text{th}}$ moment of this distribution; the first and second
moments $\Omega_{1}(\phi)$ and $\Omega_{2}(\phi)$ are the complete set of
inputs for a Gaussian distribution. Furthermore, if $\Omega_{2}$ has non-
trivial $\phi$ dependence, i.e., $\Omega_{2}^{(m\neq 0)}\neq 0$, then the
variance of the noise depends on the starting location of the jump. Finally,
if $\Omega_{n=3}(\phi)\neq 0$, then we know our distribution is non-Gaussian.
More generally, we interpret the $n>2$ terms in the generalized equation
governing Stochastic Inflation in Eq. 2.16 as encoding contributions from non-
Gaussian noise generated by the UV $\lambda\phi^{4}$ interaction. Therefore,
we conclude that corrections to Eq. 2.2 are of three types:

  * •

Higher order “noise” terms captured by $\partial^{n}/\partial\phi^{n}$ with
$n>2$, see Eq. 2.11.

  * •

Higher polynomial terms in $\Omega_{n}$, see Eq. 2.12.

  * •

Higher order terms in the potential $V$ via corrections to coefficients and
the generation of higher polynomial $\phi$ terms.

Next, we will argue for how to relate the expansion in each of these
quantities to the expansion in the UV quartic coupling $\lambda$ as it
corrects the equilibrium solution in Eq. 2.4.

####  2.2.1 Organizing Corrections Systematically

Due to the underlying $\lambda\phi^{4}$ potential, we expect that the
statistics of $\phi$ are neither Gaussian nor independent of the background
value of the field. We therefore expect corrections to the equation for
Stochastic Inflation of the form discussed previously. In this section, we
will take the equilibrium solution in Eq. 2.4 and apply it to a UV theory with

$V(\phi)=\frac{1}{4!}\lambda\hskip 0.8pt\phi^{4}\qquad\Longrightarrow\qquad
P_{\rm eq}(\phi)=Ce^{-8\pi^{2}\lambda\phi^{4}/(3\cdot 4!\,H^{4})}\equiv
Ce^{-(\phi/\phi_{\text{eq}})^{4}}\,.$ (2.17)

where $\phi^{4}_{\text{eq}}=9H^{4}/(\pi^{2}\lambda)$. This distribution has
support over a field range $|\phi|\lesssim\phi_{\text{eq}}$ such that

$\big{\langle}\phi^{4}\big{\rangle}=\frac{1}{4}\Gamma\big{[}\tfrac{5}{4}\big{]}\times\phi^{4}_{\text{eq}}=\frac{9}{4\pi^{2}}\Gamma\big{[}\tfrac{5}{4}\big{]}\frac{H^{4}}{\lambda}\,.$
(2.18)

Therefore, we will organize the possible corrections by assuming the
equilibrium scaling

$\displaystyle\phi\simeq\phi_{\text{eq}}\sim H\lambda^{-1/4}\,.$ (2.19)

We will further assume that the corrections are generated as an expansion in
perturbation theory, so that

$\displaystyle\Omega_{2}^{(2m)}\sim\lambda^{m};\qquad\Omega_{3}^{(2m+1)}\sim\lambda^{m+1};\qquad\Omega_{4}^{(2m)}\sim\lambda^{m+1};\qquad\text{and}\qquad
c_{2\ell}\sim\lambda^{\ell-1}\,,$ (2.20)

which we will see agrees with the explicit calculations presented below.
Putting all of this together allows us to determine the order in $\lambda$ for
each term that appears in Eq. 2.16. Note that we will assume the
$\phi\to-\phi$ UV symmetry is preserved, which explains the absence of many
terms. These contributions are as follows. Leading Order (LO): Stochastic
Inflation at leading order is given by Eq. 2.2:

$\frac{\partial}{\partial
t}P(\phi,t)=\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}}{\partial\phi^{2}}P(\phi,t)+\frac{1}{3H}\frac{\partial}{\partial\phi}\left[\frac{1}{3!}\lambda\phi^{3}P(\phi,t)\right]\,,$
(2.21)

where we have used the known leading order results
$\Omega_{2}^{(0)}=H^{3}/(4\pi^{2})$ and $V^{\prime}(\phi)=\lambda\phi^{3}/3!$.
Both terms on the right hand side of this equation are
${O}\big{(}\lambda^{1/2}\big{)}\times P(\phi,t)$, which defines what we mean
by “LO.”

Next-to-leading Order (NLO): Accounting for both $\Omega_{n}^{(m)}$ and
corrections to the potential, we can determine that the next-to-leading
corrections to Stochastic Inflation should take the form

$\displaystyle\frac{\partial}{\partial
t}P(\phi,t)=O\big{(}\lambda^{1/2}\big{)}+\frac{\partial^{2}}{\partial\phi^{2}}\Big{[}\Omega_{2}^{(2)}\phi^{2}P(\phi,t)\Big{]}+\frac{1}{3H}\frac{\partial}{\partial\phi}\bigg{[}\frac{1}{5!}c_{6}\phi^{5}P(\phi,t)\bigg{]}\,,$
(2.22)

where the correction to the noise term $\Omega_{2}^{(2)}\sim\lambda$ and the
correction to the potential $c_{6}\sim\lambda$ will both be determined below,
and $\Omega_{2}^{(1)}=0$ due to the $\phi\to-\phi$ symmetry. We see that these
NLO terms are $O(\lambda)\times P(\phi,t)$. These corrections have been
previously calculated in [30, 34].

Next-to-next-to-leading Order (NNLO): Following the same logic, we can find
the form that the next order terms take:

$\displaystyle\frac{\partial}{\partial t}P(\phi,t)=\,$ $\displaystyle
O\big{(}\lambda^{1/2}\big{)}+O\big{(}\lambda\big{)}+\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}}{\partial\phi^{2}}\Big{(}\Omega_{2}^{(4)}\phi^{4}P(\phi,t)\Big{)}$
$\displaystyle+\frac{1}{3H}\frac{\partial}{\partial\phi}\bigg{[}\frac{1}{7!}c_{8}\phi^{7}P(\phi,t)\bigg{]}+\frac{\partial^{3}}{\partial\phi^{3}}\Big{(}\Omega_{3}^{(1)}\phi
P(\phi,t)\Big{)}\,,$ (2.23)

where $\Omega_{2}^{(4)}\sim\lambda^{2}$ and $\Omega_{3}^{(1)}\sim\lambda$ will
be determined by operator mixing in the next section and
$c_{8}\sim\lambda^{3}$, and $\Omega_{3}^{(0)}=0$ due to the $\phi\to-\phi$
symmetry. These NNLO terms are $O(\lambda^{3/2})\times P(\phi,t)$.

Note that in addition to these corrections, we must also include subleading
corrections to the parameters that already appear at lower order; these do not
change the structure of the equation, but will of course be accounted for as
we perform the calculation. The rest of this paper is devoted to determining
these coefficients systematically using the framework of SdSET, with a brief
discussion of the physical implications of working with NNLO Stochastic
Inflation.

##  3 Soft de Sitter Effective Theory

Stochastic Inflation, and corrections to it, are the consequence of quantum
field theory in dS for scalar particles with masses $m^{2}\ll H^{2}$. This can
be seen directly at leading order, where a variety of methods have been used
to derive the Fokker-Planck equation [21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35]. However, many of these methods become cumbersome beyond
leading order and often obscure how corrections arise.

Soft de Sitter Effective Theory offers a method to compute the equations of
Stochastic Inflation systematically to any order. The key advantage offered by
SdSET is that power counting is manifest, thereby making corrections easy to
identify. Furthermore, loop integrals are scaleless and regulated by
(dynamical) dim reg and thus preserve the power counting that is manifest in
the action. In addition to fixing the values of the SdSET Wilson coefficients,
the UV theory sets the initial conditions for the effective theory fields. We
will be specifically interested in understanding the corrections to Stochastic
Inflation for the concrete example of $\lambda\phi^{4}$ theory in dS, where
the UV action is given above in Eq. 2.1.

In this section, we will review the machinery of SdSET and how it arises from
a given UV theory. In the subsequent sections, we will use this technology to
match SdSET to $\lambda\phi^{4}$ theory and then use it derive the equations
of Stochastic Inflation at NNLO.

###  3.1 In-In Correlators

As we will see below, Stochastic Inflation is equivalent to the
renormalization group equations that govern how composite operators mix. One
approach to determining the operator mixing is to compute the divergences of
in-in correlation functions involving composite operators. This section is
devoted to setting up the relevant framework. We work in the interaction
picture, where fields are quantized using the solutions to their quadratic
equations of motion. A free scalar fields in dS can be expressed as a mode
expansion:

$\phi({\vec{x}},\tau)=\int\frac{{\rm d}^{3}k}{(2\hskip 0.8pt\pi)^{3}}\hskip
0.8pte^{i\hskip 0.8pt{\vec{k}}\cdot{\vec{x}}}\Big{(}\bar{\phi}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},\tau\big{)}a_{{\vec{k}}}^{\dagger}+\bar{\phi}^{*}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},\tau\big{)}a_{-{\vec{k}}}\Big{)}\,,$ (3.1)

where $\tau=-1/[aH]$ is the conformal time, and $a_{\vec{k}}^{\dagger}$ and
$a_{\vec{k}}$ are the canonical creation and annihilation operators
respectively that satisfy

$\displaystyle\Big{[}a_{\vec{k}}^{\dagger}\hskip 0.8pt,\hskip
0.8pta_{{\vec{k}}^{\prime}}\Big{]}=(2\hskip 0.8pt\pi)^{3}\hskip
0.8pt\delta\big{(}{\vec{\scaleto{k}{7pt}}}-{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\,.$ (3.2)

In the Bunch-Davies vacuum, one finds the positive frequency modes are given
by

$\bar{\phi}\big{(}\hskip 0.8pt{\vec{\scaleto{k}{7pt}}},\tau\big{)}=-i\hskip
0.8pte^{i\hskip
0.8pt\left(\nu+\frac{1}{2}\right)\frac{\pi}{2}}\frac{\sqrt{\pi}}{2}H(-\tau)^{3/2}\text{H}_{\nu}^{(1)}(-k\hskip
0.8pt\tau)\qquad\text{with}\qquad\nu=\sqrt{\frac{9}{4}-\frac{m^{2}}{H^{2}}}\,,$
(3.3)

so that $\nu=3/2$ corresponds to a massless field (see e.g. [53] for review).
The observables of this theory are equal-time in-in correlation functions,
which are computed via

$\displaystyle\langle{\rm in}|Q(t)|{\rm in}\rangle$
$\displaystyle=\sum_{N=0}^{\infty}i^{N}\int_{-\infty}^{t}{\rm
d}t_{1}\int_{-\infty}^{t_{1}}{\rm d}t_{2}\,...\,\int_{-\infty}^{t_{N-1}}{\rm
d}t_{N}$ $\displaystyle\hskip 60.0pt\times\Big{\langle}\big{[}H_{{\rm
int}}\big{(}t_{N}\big{)},\big{[}H_{{\rm
int}}\big{(}t_{N-1}\big{)},\,...\,\big{[}H_{{\rm
int}}\big{(}t_{1}\big{)},Q_{\mathrm{int}}(t)\big{]}\,...\,\big{]}\big{]}\Big{\rangle}\,,$
(3.4)

using

$H_{\rm int}(t)=\int{\rm d}^{3}x\sqrt{-g}\hskip
0.8pt\frac{\lambda}{4!}\phi^{4}({\vec{x}},t)\,.$ (3.5)

We will be interested in multi-field correlators in the long wavelength limit,
so that
$Q(t)=\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{1},t\big{)}\,...\,\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{n},t\big{)}$.
In general, this expression must be generalized to allow for the $i\epsilon$
prescription that projects the initial state onto the interacting vacuum, but
this form is the most useful starting point for understanding the superhorizon
evolution.

For illustration, we can compute the tree-level four point correlation
function as

$\displaystyle\Big{\langle}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}_{\rm
tree}=i\frac{\lambda}{4!}\int^{t}{\rm d}t_{1}\,{\rm
d}^{3}x\,a(t)^{3}\Big{\langle}\Big{[}\phi^{4}({\vec{x}},t_{1}),\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{]}\Big{\rangle}^{\prime}\,,$
(3.6)

where the $\phi\big{(}{\vec{\scaleto{k}{7pt}}}\hskip 0.8pt\big{)}$ fields are
all evaluated at the same time $t$, and we have introduced the notation

$\displaystyle\big{\langle}\,...\,\rangle=(2\pi)^{3}\delta^{3}\big{(}\textstyle\sum{\vec{\scaleto{k}{7pt}}}_{i}\big{)}\big{\langle}\,...\,\rangle^{\prime}\,.$
(3.7)

At tree-level, we can simply use the massless mode functions,222We will use
$\frac{3}{2}-\nu\equiv\alpha\neq 0$ as a regulator when we encounter loops.

$\displaystyle\bar{\phi}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},\tau\big{)}\,\,\to\,\,\frac{H}{\sqrt{2k^{3}}}(1-ik\tau)e^{ik\tau}\,,$
(3.8)

to evaluate this expression

$\displaystyle\Big{\langle}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}_{\rm
tree}=\lambda\,2\hskip 0.8pt{\rm Im}\int^{\tau}\frac{{\rm
d}\tau_{1}}{(-H\tau_{1})^{4}}\prod_{i=1}^{4}\frac{(1-ik_{i}\tau_{1})(1+ik_{i}\tau)}{2k_{i}^{3}}e^{ik_{i}(\tau_{1}-\tau)}$
$\displaystyle=\frac{\lambda}{8(k_{1}k_{2}k_{3}k_{4})^{3}}\Bigg{[}\frac{1}{3}\Bigg{(}\sum_{i}k_{i}^{3}\Bigg{)}\bigg{(}\log\frac{k_{t}}{[aH]}+\gamma_{E}+\frac{1}{3}-2\bigg{)}-\frac{k_{1}k_{2}k_{3}k_{4}}{k_{t}}-\frac{1}{9}k_{t}^{3}$
$\displaystyle\hskip
112.0pt+2\sum_{i<j<\ell}k_{i}k_{j}k_{\ell}+\frac{1}{3}k_{t}\Bigg{(}\sum
k_{i}^{2}-\sum_{i<j}k_{i}k_{j}\Bigg{)}\Bigg{]}\,,$ (3.9)

where $k_{t}=k_{1}+k_{2}+k_{3}+k_{4}$. In the final step, we have used
$\tau=-1/[aH]$, have kept the $\log[aH]$ and time-independent contributions,
and have not included terms that vanish as $[aH]\to\infty$. In deriving this
expression, we have expressed the commutator as the imaginary part of the
integral, which holds for real fields at first order in the $H_{\rm int}$
expansion, namely

$i\Big{(}\big{\langle}H_{\rm int}(t_{1})Q(t)\big{\rangle}-\langle Q(t)H_{\rm
int}(t_{1})\rangle\Big{)}=2\hskip 0.8pt{\rm Im}\,\big{\langle}Q(t)H_{\rm
int}(t_{1})\big{\rangle}\,,$ (3.10)

for a real operator $Q(t)$.

###  3.2 Taking the Long Wavelength Limit

In this section, we review how to determine in-in correlators in the soft
limit using SdSET. The starting point is to decompose the UV fields according
to

$\displaystyle\phi_{S}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},\texttt{t}\big{)}$
$\displaystyle=H\Big{(}[a(\texttt{t})\hskip
0.8ptH]^{-\alpha}\varphi_{+}({\vec{\scaleto{k}{7pt}}},\texttt{t})+[a(\texttt{t})\hskip
0.8ptH]^{-\beta}\varphi_{-}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},\texttt{t}\big{)}\Big{)}\ .$ (3.11)

where
$\phi({\vec{x}},\texttt{t})=\phi_{S}({\vec{x}},\texttt{t})+\Phi_{H}({\vec{x}},\texttt{t})$
is split into soft (superhorizon) and hard (subhorizon) modes, and we have
introduced a dimensionless time variable $\texttt{t}\equiv Ht$ so that the
mass dimension of operators tracks the EFT power counting. The parameters
$\alpha$ and $\beta$ are subject to the constraint $\alpha+\beta=3$. This is
straightforward to derive from the top down, as these parameters are
determined by the mass via $\alpha=3/2-\nu$ and $\beta=3/2+\nu$, where $\nu$
is defined in Eq. 3.3.

The decomposition of $\phi_{S}$ into $\varphi_{+}$ and $\varphi_{-}$ is exact
in the free theory. Taking the limit $k\hskip 0.8pt\tau\ll 1$ and using
$\tau=-1/[a\hskip 0.8ptH]$ in Eq. 3.3, we find

$\displaystyle\varphi_{+}\big{(}{\vec{x}},t\big{)}$
$\displaystyle=\int\frac{{\rm d}^{3}k}{(2\hskip 0.8pt\pi)^{3}}e^{i\hskip
0.8pt{\vec{k}}\cdot\vec{x}}\hskip 0.8pt\bar{\varphi}_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},t\big{)}\tilde{a}_{{\vec{\scaleto{k}{7pt}}}}$
(3.12a) $\displaystyle\varphi_{-}\big{(}{\vec{x}},t\big{)}$
$\displaystyle=\int\frac{{\rm d}^{3}k}{(2\hskip 0.8pt\pi)^{3}}e^{i\hskip
0.8pt{\vec{k}}\cdot\vec{x}}\hskip 0.8pt\bar{\varphi}_{-}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},t\big{)}\tilde{b}_{{\vec{\scaleto{k}{7pt}}}}\ ,$
(3.12b)

where

$\displaystyle\bar{\varphi}_{+}=C_{\alpha}\frac{1}{\sqrt{2}\hskip
0.8ptk^{\frac{3}{2}-\alpha}}\,,\qquad\text{and}\qquad\bar{\varphi}_{-}=D_{\beta}\frac{1}{\sqrt{2}\hskip
0.8ptk^{\frac{3}{2}-\beta}}\,,$ (3.13)

and

$\displaystyle C_{\alpha}$ $\displaystyle=2^{1-\alpha}\hskip
0.8pt\frac{\Gamma\big{(}\frac{3}{2}-\alpha\big{)}}{\sqrt{\pi}}\,,\qquad\text{and}\qquad
D_{\beta}=-2^{1-\beta}\hskip 0.8pt\frac{\sqrt{\pi}}{\cos(\pi\hskip
0.8pt\beta)\hskip 0.8pt\Gamma\big{(}\beta-\frac{1}{2}\big{)}}\ .$ (3.14)

The operators $\tilde{a}_{{\vec{\scaleto{k}{7pt}}}}$ and
$\tilde{b}_{{\vec{\scaleto{k}{7pt}}}}$ are given in terms of the UV creation
and annihilation operators of the form

$\displaystyle\tilde{a}_{\vec{k}}=e^{i\hskip
0.8pt\delta_{\nu}}a^{\dagger}_{\vec{k}}+e^{-i\hskip
0.8pt\delta_{\nu}}a_{-{\vec{k}}}\,,\qquad\text{and}\qquad\tilde{b}_{\vec{k}}=i\Big{(}e^{-i\hskip
0.8pt\delta_{\nu}}a^{\dagger}_{\vec{k}}-e^{i\hskip
0.8pt\delta_{\nu}}a_{-{\vec{k}}}\Big{)}\,.$ (3.15)

From the UV theory, we determine that the operators commute with themselves

$\displaystyle\Big{[}\tilde{a}_{\vec{k}}^{\dagger}\hskip 0.8pt,\hskip
0.8pt\tilde{a}_{\vec{k}}\Big{]}=\Big{[}\tilde{b}_{\vec{k}}^{\dagger}\hskip
0.8pt,\hskip 0.8pt\tilde{b}_{\vec{k}}\Big{]}=0\ .$ (3.16)

Nevertheless, these operators still have non-zero correlation function

$\displaystyle\Big{\langle}\tilde{a}_{\vec{k}}\,\tilde{a}_{{\vec{k}}^{\prime}}\Big{\rangle}$
$\displaystyle=(2\pi)^{3}\hskip
0.8pt\delta\big{(}{\vec{\scaleto{k}{7pt}}}+{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}$ (3.17a)
$\displaystyle\Big{\langle}\tilde{b}_{\vec{k}}\,\tilde{b}_{{\vec{k}}^{\prime}}\Big{\rangle}$
$\displaystyle=(2\pi)^{3}\hskip
0.8pt\delta\big{(}{\vec{\scaleto{k}{7pt}}}+{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\ ,$ (3.17b)

where $\langle..\rangle\equiv\langle 0|..|0\rangle$, and $|0\rangle$ is the
vacuum that is annihilated by $a_{\vec{k}}$. This gives rise to classical
statistical power spectra

$\displaystyle\Big{\langle}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\,\varphi_{+}\big{(}{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\hskip 0.8pt\big{)}\Big{\rangle}$
$\displaystyle=\frac{C_{\alpha}^{2}}{2}\frac{1}{k^{3-2\hskip
0.8pt\alpha}}\hskip 0.8pt(2\hskip 0.8pt\pi)^{3}\hskip
0.8pt\delta\big{(}{\vec{\scaleto{k}{7pt}}}+{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}$ (3.18a)
$\displaystyle\Big{\langle}\varphi_{-}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\,\varphi_{-}\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\Big{\rangle}$
$\displaystyle=\frac{D_{\beta}^{2}}{2}\frac{1}{k^{3-2\hskip 0.8pt\beta}}\hskip
0.8pt(2\hskip 0.8pt\pi)^{3}\hskip
0.8pt\delta\big{(}{\vec{\scaleto{k}{7pt}}}+{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\,,$ (3.18b)

where $C_{\alpha}$ and $D_{\beta}$ are defined in Eq. 3.14 above. Note that in
the massless limit $\alpha\to 0$, we reproduce the famous scale invariant
power spectrum.

The corrections to this mapping can be systematically accounted for by
matching between the UV theory and the EFT, see Sec. 4. The fields
$\varphi_{+}$ and $\varphi_{-}$ have well defined power counting; they carry
operator dimension $\alpha$ and $\beta$ respectively. After utilizing field
redefinitions, on-shell conditions, and power counting to remove redundant
operators, the low energy effective action is given by

$\displaystyle S_{\pm}=\int{\rm d}^{3}x\,{\rm d}\texttt{t}\hskip
0.8pt\Bigg{[}-\nu\hskip 0.8pt\big{(}\dot{\varphi}_{+}\hskip
0.8pt\varphi_{-}-\varphi_{+}\hskip 0.8pt\dot{\varphi}_{-}\big{)}-\sum_{n\geq
2}^{\infty}\hskip 0.8pt[a\hskip 0.8ptH]^{3-n\hskip 0.8pt\alpha-\beta}\hskip
0.8pt\frac{c_{n,1}}{n!}\hskip 0.8pt\varphi_{+}^{n}\hskip
0.8pt\varphi_{-}\Bigg{]}\,,$ (3.19)

where the $c_{n,1}$ are dimensionless Wilson coefficients. Note that t carries
dimension zero by SdSET power counting, so marginal operators are dimension
three. This explains why we have only included operators with a single factor
of $\varphi_{-}$ since these are the only terms that become marginal in the
massless limit ($\alpha\to 0$). Additionally, we have not included any terms
with $\vec{\partial}$, which start at dimension five and are therefore power
suppressed by at least $k^{2}/[aH]^{2}$.

In addition to an action, SdSET requires specifying _initial conditions_ for
the fields $\varphi_{+}$ and $\varphi_{-}$ that are acquired from the time
evolution prior to horizon crossing. These initial conditions are random such
that, to leading order, $\varphi_{+}$ behaves as a classical stochastic
variable with correlations fixed by matching

$\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{N}\big{)}\Big{\rangle}_{{\rm
IC}^{(n)}}=\mathcal{K}^{-3\hskip 0.8pt(N-1)+N\hskip
0.8pt\alpha}F_{(n)}\big{(}\\{{\vec{q}}_{i}\\}\big{)}\hskip 0.8pt(2\hskip
0.8pt\pi)^{3}\hskip
0.8pt\delta\Big{(}\sum{\vec{\scaleto{k}{7pt}}}_{i}\Big{)}\,,$ (3.20)

where $\mathcal{K}$ is a reference momentum scale,
$F_{(n)}\big{(}\\{{\vec{q}}_{i}\\}\big{)}$ encode the dependence on the
rescaled momenta $\vec{q}_{i}=\vec{k}_{i}/\mathcal{K}$, and the $(n)$
subscripts track the order in the $\lambda$ perturbative expansion for each
contribution; the two point correlators in Eq. 3.18 should be viewed as
$\langle\,...\,\rangle_{{\rm IC}^{(0)}}$. Because $\varphi_{+}$ is time
independent to leading order in the EFT, the initial conditions are determined
by matching the time independent terms.

We evaluate time integrals in the EFT using

$\int^{\texttt{t}}_{-\infty}{\rm
d}\texttt{t}^{\prime}\big{[}a\big{(}\texttt{t}^{\prime}\big{)}H\big{]}^{\gamma}=\frac{1}{\gamma}[a(\texttt{t})H]\,,$
(3.21)

where we assume that this holds even when $\gamma<0$. This analytic
continuation enforces that the contributions from early times vanish, thereby
ensuring that power law divergences associated with physics at horizon
crossing are automatically absorbed into the initial conditions (in close
analogy with how dim reg treats power law divergences).

Cosmological correlators are determined in SdSET using the same in-in
formalism as applied to the UV theory, see Sec. 3.1. For illustration, we can
compute the tree-level trispectrum using Eq. 3.4 and the canonical commutator

$\Big{[}\varphi_{+}\big{(}{\vec{x}},\texttt{t}\big{)}\hskip 0.8pt,\hskip
0.8pt\varphi_{-}\big{(}{\vec{x}}^{\hskip
0.8pt\prime},\texttt{t}\big{)}\Big{]}=-\frac{i}{2\hskip 0.8pt\nu}\hskip
0.8pt\delta\big{(}{\vec{x}}-{\vec{x}}^{\hskip 0.8pt\prime}\big{)}\,.$ (3.22)

Performing the time integrals using Eq. 3.21, we find

$\displaystyle\Big{\langle}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\hskip
0.8pt\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\hskip
0.8pt\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\hskip
0.8pt\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}=\bigg{\langle}\bigg{[}\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{1})\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{2})\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{3})\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{4})\,,$
$\displaystyle\hskip 160.0pt(-i)\frac{c_{3,1}}{3!}\int{\rm
d}\texttt{t}^{\prime}\,{\rm
d}^{3}x^{\prime}\,\big{[}a\big{(}\texttt{t}^{\prime}\big{)}H\big{]}^{-2\alpha}\varphi_{+}^{3}\hskip
0.8pt\varphi_{-}\big{(}{\vec{x}}\hskip
0.8pt^{\prime},\texttt{t}^{\prime}\big{)}\bigg{]}\bigg{\rangle}$
$\displaystyle=-\frac{c_{3,1}}{2\nu}\bigg{\langle}\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{2})\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{3})\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{4})\int{\rm
d}^{3}x^{\prime}\varphi_{+}^{3}\hskip 0.8pt\bigg{\rangle}\int{\rm
d}\texttt{t}^{\prime}\big{[}a\big{(}\texttt{t}^{\prime}\big{)}H\big{]}^{-2\alpha}+{\rm
permutations}$
$\displaystyle=\frac{c_{3,1}}{2\nu}\frac{C_{\alpha}^{6}\sum_{i}k_{i}^{3-2\alpha}}{\big{(}k_{1}k_{2}k_{3}k_{4}\big{)}^{3-2\alpha}}\bigg{(}\frac{[aH]^{-2\alpha}}{2\alpha}\bigg{)}\,,$
(3.23)

where $c_{3,1}$ is the Wilson coefficient for the $\varphi_{+}^{3}\varphi_{-}$
operator, see Eq. 3.19, and we used Eq. 3.18a to evaluate the field
contractions. In addition, we must include any trispectrum associated with the
initial conditions.

###  3.3 (Dynamical) Renormalization

Loops corrections are calculated in the SdSET using dynamical dimensional
regularization (dynamical dim reg). Rather than varying the spacetime
dimension, we instead float the dynamical exponents $\alpha$, and evaluate
loop integrals by analytic continuation in $\alpha$. Then when we encounter
divergences as $\alpha\to 0$, they will be accompanied by log corrections to
the time evolution, in exact analogy with conventional dim reg. To keep the
units fixed as we vary $\alpha$, we will introduce the necessary powers of
$[aH]$ such that $\varphi_{+}$ stays dimensionless. Then we take $\alpha=0$ at
the end of the calculation, so that:

$\displaystyle\phi_{S}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},\texttt{t}\big{)}\to
H\Big{(}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}},\texttt{t}\big{)}+[a(\texttt{t})\hskip
0.8ptH]^{-3}\varphi_{-}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}},\texttt{t}\big{)}\Big{)}\,,$ (3.24)

so that $\varphi_{+}$ corresponds to a massless mode.

Since we are working within the EFT, we will typically encounter vanishing
scaleless integrals. Then we can isolate the UV divergence in the usual way by
regulating the IR with a dimensionful parameter $K$:

$\displaystyle\langle{\cal O}\,...\,\rangle$
$\displaystyle\propto[aH]^{-2\alpha}\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\frac{1}{p^{3-2\alpha}}$
$\displaystyle\to[aH]^{-2\alpha}\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\frac{1}{(p^{2}+K^{2})^{3/2-\alpha}}=[aH]^{-2\alpha}\frac{1}{8\pi^{3/2}}\frac{\Gamma[-\alpha]}{\Gamma[3/2-\alpha]}K^{2\alpha}$
$\displaystyle\to-\frac{1}{2\pi^{2}}\left(\frac{1}{2\alpha}+\log\frac{K}{[aH]}-\log
2\right)\,,$ (3.25)

where we have taken the limit $\alpha\to 0$ in the third line. Having isolated
these UV divergent contributions, we can use them to determine the (dynamical)
RG flow, which resums a series in $\log[aH]$.

Having regulated the divergence, we can then absorb it into the
renormalization of the operator

${\cal O}=Z_{\cal O}\hskip 0.8pt{\cal O}_{R}\qquad\text{with}\qquad Z_{\cal
O}-1\propto-\frac{1}{2\pi^{2}}\left(\frac{1}{2\alpha}+\log\frac{K_{\star}}{[aH]_{\star}}\right)\,,$
(3.26)

so that

$\langle{\cal O}_{R}\,...\,\rangle\propto\frac{1}{2\pi^{2}}\log\frac{\hskip
2.0pt[aH]_{\star}}{[aH]}+\log\frac{K}{\hskip 1.0ptK_{\star}}\,,$ (3.27)

where $[aH]_{\star}$ and $K_{\star}$ are energy and momentum scales we have
invented to make the logs small, i.e., subtraction points. By definition the
bare operator ${\cal O}$ is independent of these arbitrary parameters, and
thus

$\frac{{\rm d}}{{\rm d}\log[aH]_{\star}}\hskip 0.8pt{\cal
O}=0\quad\Longleftrightarrow\quad\frac{{\rm d}}{{\rm d}\log[aH]_{\star}}\hskip
0.8pt{\cal O}_{R}=-\frac{{\rm d}\log Z_{\cal O}}{{\rm
d}\log[aH]_{\star}}\hskip 0.8pt{\cal O}_{R}\equiv\gamma\hskip 0.8pt{\cal
O}_{R}\,,$ (3.28)

where the anomalous dimension $\gamma$ is independent of $[aH]$. We are
working with a scheme where $Z_{\cal O}$ is diagonal at the scale
$[aH]_{\star}$. In general, $\gamma\to\gamma_{ij}$ is a matrix that acts on
the space of operators, and which encodes both the anomalous scaling and
mixing of these operators.

###  3.4 Matching and Initial Conditions

SdSET provides an effective description for the time evolution of scalar modes
that have crossed the Hubble horizon. Their state at the time of horizon
crossing cannot be computed within the EFT, and instead has to be provided as
an additional input. This is why SdSET requires matching for both the initial
conditions and the EFT Wilson coefficients. This is not unique to SdSET, but
is necessarily part of any EFT description of the post inflationary universe
as well, see e.g. [54].

When defined using dynamical dim reg, SdSET is a so-called continuum EFT [55].
Concretely, the time integrals include arbitrary early times, even though the
EFT does not provide a model of the subhorizon physics, since it relies on the
use of the long-wavelength mode functions at all times. Importantly, as with
all continuum EFT, these early time integrals only make scaleless (and
therefore vanishing) contributions. Thus we can integrate over all times, such
that the regulated integrals respect the low energy symmetries and the EFT
power counting. Underlying the validity of this procedure is the fact that the
subhorizon physics only alters the initial conditions and thus we can fully
account for all subhorizon evolution by matching a UV theory onto SdSET.

One can demonstrate how matching separates into Wilson coefficient and initial
condition corrections more directly using a hard cutoff to evaluate time and
momentum integrals, i.e., treating the theory as a Wilsonian EFT [55]. Let
$k_{i}$ denote the magnitudes of the momenta appearing in the cosmological
correlator, and let the cutoff of the momentum integrals $\Lambda$ be much
greater than any of the $k_{i}$. In addition, we denote the time cutoff by
$t_{\Lambda}$; this is the time when all the EFT modes are in the superhorizon
limit, and thus it specifies when we set the initial conditions for the EFT.

Before $t_{\Lambda}$, a subset of EFT modes are subhorizon, and so one must
use the UV theory to describe them. To match onto the EFT, it is useful to
split the full theory time evolution into pieces before and after
$t_{\Lambda}$. Let $U_{I}(t,t^{\prime})$ represent the interaction picture
propagator, and $|\Omega\rangle$ be the UV vacuum state. This decomposition
can then be written as

$\displaystyle\big{\langle}\Omega\big{|}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{1},t\big{)}\,...\,\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{n},t\big{)}\big{|}\Omega\big{\rangle}=\big{\langle}\Omega\big{|}U_{I}^{\dagger}(t,-\infty)\phi_{I}\big{(}{\vec{\scaleto{k}{7pt}}}_{1},t\big{)}\,...\,\phi_{I}\big{(}{\vec{\scaleto{k}{7pt}}}_{n},t\big{)}U_{I}(t,-\infty)\big{|}\Omega\big{\rangle}$
$\displaystyle\hskip
29.0pt=\big{\langle}\Omega\big{|}U_{I}^{\dagger}(t_{\Lambda},-\infty)U_{I}^{\dagger}(t,t_{\Lambda})\phi_{I}\big{(}{\vec{\scaleto{k}{7pt}}}_{1},t\big{)}\,...\,\phi_{I}\big{(}{\vec{\scaleto{k}{7pt}}}_{n},t\big{)}U_{I}(t,t_{\Lambda})U_{I}(t_{\Lambda},-\infty)\big{|}\Omega\big{\rangle}\,,$
(3.29)

where $\phi_{I}$ is the interaction picture field.

One can trivially re-write Eq. 3.29 as an expectation value of
$\ket{\psi}=U_{I}(t_{\Lambda},-\infty)\ket{\Omega}$, the state of the full
theory fields at $t_{\Lambda}$. One can then integrate out the modes whose
wave vector magnitudes satisfy $k>\Lambda$, so that the remaining modes are
superhorizon after $t_{\Lambda}$. It was shown in [33] that, after integrating
out these so-called hard modes, the resulting action for the superhorizon
modes $\varphi_{+}$ and $\varphi_{-}$ is local. We can then evolve these modes
from $t_{\Lambda}$ to $t$ using the unitary time evolution operators defined
within the SdSET itself:

$\displaystyle\big{\langle}\Omega\big{|}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{1},t\big{)}\,...\,\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{n},t\big{)}\big{|}\Omega\big{\rangle}$
$\displaystyle=$ $\displaystyle\big{\langle}\psi_{\rm
EFT}(t_{\Lambda})\big{|}U_{I,{\rm EFT}}^{\dagger}$
$\displaystyle(t,t_{\Lambda})\phi_{S}\big{(}{\vec{\scaleto{k}{7pt}}}_{1},t\big{)}\,...\,\phi_{S}\big{(}{\vec{\scaleto{k}{7pt}}}_{n},t\big{)}U_{I,{\rm
EFT}}(t,t_{\Lambda})\big{|}\psi_{\rm EFT}(t_{\Lambda})\big{\rangle}\,,$ (3.30)

where $U_{I,{\rm EFT}}$ is the interaction picture propagator obtained from
Eq. 3.19, and $\phi_{S}$ is given by Eq. 3.24. The state $|\psi_{\rm
EFT}(t_{\Lambda})\rangle$ is the EFT state inherited from $|\psi\rangle$ that
results from integrating out the hard modes; this state encodes the initial
conditions for $\varphi_{+}$ and $\varphi_{-}$.

One then fixes the EFT parameters in Eq. 3.19 and the initial state
$\ket{\psi_{\rm EFT}(t_{\Lambda})}$ by matching to full theory correlators. In
practice, the full form of $\ket{\psi_{\rm EFT}(t_{\Lambda})}$ is more
information than is needed to derive the late time behavior of
$n\text{-point}$ correlation functions of $\phi$. Instead, it is sufficient to
determine a finite number of $n$-point functions,

$\displaystyle\big{\langle}\psi_{\rm
EFT}(t_{\Lambda})\big{|}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{n}\big{)}\big{|}\psi_{\rm
EFT}(t_{\Lambda})\big{\rangle}\,,$ (3.31)

from matching. Furthermore, this shows that all of the contributions from
$t<t_{\Lambda}$ are encode in the state $\ket{\psi_{\rm EFT}(t_{\Lambda})}$
or, equivalently, in the initial conditions set at $t_{\Lambda}$. Finally,
since $t_{\Lambda}$ is an unphysical cutoff parameter, no physical results can
depend on it. In particular, the initial conditions can be identified as the
_time-independent_ contribution from the UV correlators. This implies that we
do not need to rely on hard cutoff to derive the initial conditions. In what
follows, we will regulate the theory using dynamical dim reg, and will derive
the initial conditions by identifying the time-independent contributions to
correlators that appear when matching.

To illustrate the matching procedure, it is useful to first consider a simple
example. By expanding Eq. 3.4 to leading order in $c_{3,1}$ and setting
$\ket{\rm in}=\ket{\psi_{\rm EFT}}$ we find that the tree-level EFT prediction
for the four point function of $\phi$ is

$\displaystyle\big{\langle}\Omega\big{|}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{1},t\big{)}\,...\,\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{4},t\big{)}\big{|}\Omega\big{\rangle}=\left(\frac{\tau}{\tau_{\Lambda}}\right)^{4\alpha}\bigg{[}\big{\langle}\psi_{\rm
EFT}(t_{\Lambda})\big{|}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\big{|}\psi_{\rm
EFT}(t_{\Lambda})\big{\rangle}$ $\displaystyle\hskip
6.0pt+2ic_{3,1}\int_{\tau_{\Lambda}}^{\tau}\frac{{\rm
d}\tau_{1}}{\tau_{1}^{1+2\alpha}}\int{\rm d}^{3}x\,\big{\langle}\psi_{\rm
EFT}(t_{\Lambda})\big{|}\big{[}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)},\varphi_{+}^{3}\varphi_{-}({\vec{x}},\tau_{1})\big{]}\big{|}\psi_{\rm
EFT}(t_{\Lambda})\big{\rangle}\bigg{]}\,.$ (3.32)

As in all perturbative SdSET calculations, the time dependence and field
contractions factorize at the integrand level. The EFT prediction is then
parameterized by the coupling $c_{3,1}$ and two expectations values of
$\ket{\psi_{\rm EFT}}$.

The expectation values parameterize the subhorizon evolution of the EFT modes
before $t_{\Lambda}$. If the full theory is perturbative, then the subhorizon
evolution is approximately Gaussian. Assuming the UV theory is given by
$\lambda\phi^{4}$, then to $O(\lambda)$ we can write

$\displaystyle\big{\langle}\psi_{\rm
EFT}(t_{\Lambda})\big{|}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\big{|}\psi_{\rm
EFT}(t_{\Lambda})\big{\rangle}=\Big{(}\big{\langle}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\big{\rangle}_{{\rm
IC}^{(0)}}\big{\langle}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\big{\rangle}_{{\rm
IC}^{(0)}}$
$\displaystyle+2\big{\langle}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\big{\rangle}_{{\rm
IC}^{(0)}}\big{\langle}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\big{\rangle}_{{\rm
IC}^{(1)}}+{\rm
perms}\Big{)}+\big{\langle}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\big{\rangle}_{{\rm
IC}^{(1)}}\,,$

where the final term on the RHS encodes the non-Gaussian contribution to the
subhorizon evolution of the modes. There is a similar formula for the
expectation value on the second line of Eq. 3.32. However, since it is already
multiplied by $c_{3,1}$, we only need the
$\langle...\rangle_{\text{IC}^{(0)}}$ contribution to the expectation value at
this order.

The SdSET two point functions derived from the free theory are given in Eq.
3.18. To include the impact of the UV interaction on the EFT, we compute
$\big{\langle}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\big{\rangle}_{\rm
IC}$ and $c_{3,1}$ by matching to the full theory. The superhorizon evolution
between $t_{\Lambda}$ and $t$ generates the term proportional to $c_{3,1}$,
and so we can isolate the initial conditions contribution by evaluating both
sides of Eq. 3.32 at $t_{\Lambda}$, giving

$\displaystyle\big{\langle}\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{I,+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\big{\rangle}_{\rm
IC}=\big{\langle}\Omega\big{|}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{1},t_{\Lambda}\big{)}\,...\,\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{4},t_{\Lambda}\big{)}\big{|}\Omega\big{\rangle}_{\rm
connected}\,,$ (3.34)

where the “connected” subscript refers to the fact that this does not include
the contributions from products of lower point contractions. The RHS of Eq.
3.34 can be computed using Eq. 3.4, where $H_{I}$ is given by the full theory
interaction Hamiltonian and the time integrals extend from $-\infty$ to
$t_{\Lambda}$. Since the integral’s main region of support occurs when the
modes are subhorizon, we cannot replace the mode functions with their late
time behavior, and instead have to use their full UV form given in Eq. 3.3.
The UV mode functions simplify for scalars whose mass is much lighter than the
Hubble constant and can be approximated by Eq. 3.8.

One can then fix $c_{3,1}$ by demanding that Eq. 3.32 reproduces the full
theory prediction for the correlator in the regime $t>t_{\Lambda}$. While the
split between subhorizon ($t<t_{\Lambda}$) and superhorizon ($t>t_{\Lambda}$)
evolution is manifest in the EFT, this split has to be inputted by hand in the
full theory, as was done in Eq. 3.29; this is effectively making a choice of
scheme. Practically, one can decompose the time integrals in Eq. 3.4 into
regions before and after $t_{\Lambda}$. In our four point example, subhorizon
contribution is already taken care of by the initial conditions, i.e., Eq.
3.34, while the second term in Eq. 3.32 must reproduce the superhorizon
evolution. The Wilson coefficient $c_{3,1}$ is fixed by this condition.

Since the split time $t_{\Lambda}$ is arbitrary from the perspective of the
full theory, all full-theory and EFT predictions of the $\phi$ correlators
must be independent of it. This provides an additional check on the matching
calculation in the hard cutoff scheme. Fortunately, when we calculate with
dynamical dim reg, the time integrals are manifestly independent of
$t_{\Lambda}$, while still maintaining the split between initial conditions
and time-evolution. In this sense, the hard-cutoff scheme proves the validity
of dynamical dim reg, while dynamical dim reg makes it manifest that we can
implement this procedure without breaking symmetries. This matches the more
intuitive argument that our treatment of initial conditions and time evolution
in SdSET is identical to the continuum EFT approach.

###  3.5 Stochastic Inflation from SdSET

From the point of view of our EFT, Stochastic Inflation can be understood as a
consequence of operator mixing. Specifically, for light fields for which
$\alpha\to 0$, the composite operators $\varphi_{+}^{n}$ are degenerate to
leading order (in that they have the same dimension as determined by the EFT
power counting). Assuming the correlations of these fields are only due to the
Gaussian contribution given in Eq. 3.18, one encounters a UV divergence from a
one-loop contraction

$\displaystyle\big{\langle}\varphi_{+}^{n}({\vec{x}})\,...\,\big{\rangle}$
$\displaystyle\supset\big{\langle}\varphi_{+}^{n-2}({\vec{x}})\,...\,\big{\rangle}\times\binom{n}{2}\hskip
0.8pt\frac{C_{\alpha}^{2}}{2}\int\frac{{\rm d}^{3}p}{(2\hskip
0.8pt\pi)^{3}}\frac{H^{2-2\hskip 0.8pt\alpha}}{p^{3-2\hskip 0.8pt\alpha}}$
$\displaystyle\supset\big{\langle}\varphi_{+}^{n-2}({\vec{x}})\,...\,\big{\rangle}\times\binom{n}{2}\hskip
0.8pt\frac{C_{\alpha}^{2}}{4\pi^{2}}\log[aH]\,.$ (3.35)

The dynamical RG associated with this operator mixing can be written as

$\frac{\partial}{\partial\texttt{t}}\big{\langle}\varphi_{+}^{n}({\vec{x}})\,...\,\big{\rangle}=\frac{n(n-1)}{8\hskip
0.8pt\pi^{2}}\hskip
0.8pt\big{\langle}\varphi_{+}^{n-2}({\vec{x}})\,...\,\big{\rangle}-\frac{n}{3}\sum_{m>1}\frac{c_{m,1}}{m!}\big{\langle}\varphi_{+}^{n-1}({\vec{x}})\hskip
0.8pt\varphi_{+}^{m}({\vec{x}})\,...\,\big{\rangle}\,,$ (3.36)

where the second term arises from the classical time evolution. This equation
contains the same information as Starobinsky’s formulation of Stochastic
Inflation. Specifically, we can use the Fokker-Planck equation given in Eq.
2.2 to see that these two approaches are equivalent:

$\displaystyle\frac{\partial}{\partial\texttt{t}}\big{\langle}\varphi_{+}^{n}\big{\rangle}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial\texttt{t}}\int{\rm
d}\varphi_{+}\varphi_{+}^{n}P(\varphi_{+},\texttt{t})$ (3.37) $\displaystyle=$
$\displaystyle\int{\rm
d}\varphi_{+}\varphi_{+}^{n}\left(\frac{1}{8\pi^{2}}\frac{\partial^{2}}{\partial\varphi_{+}^{2}}P(\varphi_{+},\texttt{t})+\frac{1}{3}\frac{\partial}{\partial\varphi_{+}}\left[\sum_{m>1}\frac{c_{m,1}}{m!}\varphi_{+}^{m}P(\varphi_{+},\texttt{t})\right]\right)$
$\displaystyle=$ $\displaystyle\int{\rm
d}\varphi_{+}\left(\frac{n(n-1)}{8\pi^{2}}\varphi_{+}^{n-2}P(\varphi_{+},\texttt{t})-\frac{n}{3}\varphi_{+}^{n-1}\sum_{m>1}\frac{c_{m,1}}{m!}\varphi_{+}^{m}P(\varphi_{+},\texttt{t})\right)$
$\displaystyle=$
$\displaystyle\frac{n(n-1)}{8\pi^{2}}\big{\langle}\varphi_{+}^{n-2}\big{\rangle}-\frac{n}{3}\sum_{m>1}\frac{c_{m,1}}{m!}\big{\langle}\varphi_{+}^{n-1}\varphi_{+}^{m}\big{\rangle}\,,$

where in the second line we plugged in Eq. 2.2, $Ht=\texttt{t}$, the
$\alpha=0$ relation $\phi|_{\varphi_{-}=0}\to H\varphi_{+}$, and

$V^{\prime}(\phi)\,\,\to\,\,\frac{\partial}{\partial\varphi_{-}}V(\varphi_{+},\varphi_{-})\Big{|}_{\varphi_{-}=0}=\sum_{m>1}\frac{c_{m,1}}{m!}\varphi_{+}^{m}\,.$
(3.38)

This shows that Eq. 3.36 is equivalent to the leading order equation for
Stochastic Inflation Eq. 2.2. Therefore, calculating corrections to Stochastic
Inflation has been reduced to the straightforward task of computing the higher
order dynamical RG equation using SdSET. Concretely, we would expect to find
mixing between composite operators $\varphi_{+}^{n}$ and all possible
$\varphi_{+}^{n^{\prime}}$ such that

$\displaystyle\frac{\partial}{\partial\texttt{t}}\big{\langle}\varphi_{+}^{n}\big{\rangle}=$
$\displaystyle-\frac{n}{3}\sum_{m>1}^{\text{odd}}\frac{c_{m,1}}{m!}\big{\langle}\varphi_{+}^{n+m-1}\big{\rangle}+\binom{n}{2}\sum_{m=0}^{\infty}b_{m}\big{\langle}\varphi_{+}^{n+2m-2}\big{\rangle}$
$\displaystyle-\binom{n}{3}\sum_{m=0}^{\infty}d_{m}\big{\langle}\varphi_{+}^{n+2m-2}\big{\rangle}+\binom{n}{4}\sum_{m=0}^{\infty}e_{m}\big{\langle}\varphi_{+}^{n+2m-4}\big{\rangle}+...\
.$ (3.39)

Note the role of the binomial coefficient which will originate from the number
of fields inside $\varphi_{+}^{n}$ whose contractions are responsible for
mixing with a given operator, leading to a single log divergence. Repeating
the above argument, we see that this dynamical RG is equivalent to

$\displaystyle\frac{\partial}{\partial\texttt{t}}P(\varphi_{+},\texttt{t})$
$\displaystyle=\frac{1}{3}\frac{\partial}{\partial\varphi_{+}}\left[\partial_{\varphi_{-}}V(\varphi_{+},\varphi_{-})|_{\varphi_{-}=0}P(\varphi_{+},\texttt{t})\right]+\frac{\partial^{2}}{\partial\varphi_{+}^{2}}\left[\sum_{m=0}^{\infty}\frac{b_{m}}{2!}\varphi_{+}^{2m}P(\varphi_{+},\texttt{t})\right]$
$\displaystyle\hskip
14.0pt+\frac{\partial^{3}}{\partial\varphi_{+}^{3}}\left(\varphi_{+}\sum_{m=0}^{\infty}\frac{d_{m}}{3!}\varphi_{+}^{2m}P(\varphi_{+},\texttt{t})\right)+\frac{\partial^{4}}{\partial\varphi_{+}^{4}}\left(\sum_{m=0}^{\infty}\frac{e_{m}}{4!}\varphi_{+}^{2m}P(\varphi_{+},\texttt{t})\right)+\ldots\
,$

where we see that number of derivatives is related to the binomial coefficient
of the associated mixing term. Comparing with Sec. 2.2, we see that the NLO
corrections are determined by $c_{5,1}$ and $b_{1}$ while the NNLO
coefficients are $c_{7,1}$, $b_{2}$ and $d_{0}$. Achieving NNLO accuracy
requires matching $\lambda\phi^{4}$ theory onto the SdSET at one loop, the
subject of the next section. Note, however, that we have described Stochastic
Inflation in terms of $\varphi_{+}$ rather than the UV field $\phi$. This
distinction will be important because the equations of Stochastic Inflation
are not invariant under field redefintions. We will address these issues in
detail in Sec. 5.3. Finally, we note that this result demonstrates the
Markovian assumption which led to Eq. 2.16 does in fact hold as a consequence
of SdSET power counting. Specifically, the fact that the $\varphi_{-}$
dynamics are irrelevant to the evolution of the $\varphi_{+}$ correlators
implies that the evolution of the system is indeed linear, and thus it has no
“memory.”

##  4 Matching $\lambda\phi^{4}$ Onto SdSET at One-Loop

In this section we will show how to match correlators of $\phi$ to correlators
of $\varphi_{+}$ in the SdSET to determine the EFT parameters in terms of UV
data. This then serves as input for any calculation of cosmological
correlators for $\lambda\phi^{4}$ theory in the long wavelength limit,
including the corrections to Stochastic Inflation we will discuss in
subsequent sections. Furthermore, by extending this program to one-loop order
for the first time, this calculation will serve as a non-trivial check of the
SdSET framework.

Unlike conventional EFTs, we match both the couplings of SdSET and the
stochastic initial conditions, the former results from studying the time-
dependent terms in the EFT while the latter is fixed at the time of horizon
crossing. As a result, consistent matching of time-dependence of the UV
correlators is non-trivial and requires that the SdSET is a complete
representation of the long wavelength dynamics. In contrast, time-independent
contributions to a given correlator can always be absorbed into the initial
conditions (up to composite operators as discussed in Sec. 4.3). As a result,
for tree and one-loop matching we will be particularly focused on time-
dependent UV contributions.

###  4.1 Tree-level Matching and Field Redefinitions

Tree-level matching in the interacting theory is non-trivial due to the impact
on the initial conditions. We will need to introduce non-Gaussian initial
conditions in order to match higher point correlation functions as calculated
by the UV theory.

We can understand many important aspects of matching by Taylor expanding the
UV calculation in the long wavelength limit. As a simple demonstration, we can
explore the superhorizon behavior of the operator $\phi$. At first order in
the coupling, we can apply the definition of the in-in correlator given in Eq.
3.4 with $Q(t)=\phi$, which gives

$\displaystyle[\phi]_{\lambda}\big{(}{\vec{x}},\tau\big{)}$
$\displaystyle=i\int^{\tau}{\rm d}\tau_{1}\Big{[}H_{\rm
int}(\tau_{1}),\phi_{\rm in}\big{(}{\vec{x}},\tau\big{)}\Big{]}$
$\displaystyle=\frac{\lambda}{3!}\int^{\tau}\frac{{\rm
d}\tau_{1}}{(-H\tau_{1})^{4}}\int{\rm d}^{3}x_{1}\,\phi_{\rm
int}({\vec{x}}_{1},\tau_{1})^{3}\,i\Big{[}\phi_{\rm
int}\big{(}{\vec{x}}_{1},\tau_{1}\big{)},\phi_{\rm
int}\big{(}{\vec{x}},\tau\big{)}\Big{]}\ ,$ (4.1)

where $\phi_{\rm int}$ are the interaction picture fields. Because the time
integral runs over all times, this includes both the regime where the modes
are hard (UV) and the long wavelength limit where the EFT applies.
Nevertheless, if we expand in the long wavelength limit and evaluate the
integrals with dynamical dim reg, we will only get contributions from late
times. Since $\phi_{\rm in}$ are the free field operators, we can use the map
given in Eq. 3.11 to determine the long wavelength behavior of the full
theory:333Note we are not working in the EFT yet, because we have not
integrated out the hard modes. All we are doing here is taking the long
wavelength limit.

$\displaystyle\phi_{\rm int}$ $\displaystyle\to
H\big{(}[aH]^{-\alpha}\varphi_{+}+[aH]^{-3+\alpha}\varphi_{-}\big{)}\,.$
(4.2a)

Then using $\tau=-1/[aH]$ and keeping only terms that survive as
$[aH]\to\infty$, one finds

$\displaystyle[\phi]_{\lambda}\big{(}{\vec{x}},\texttt{t}\big{)}$
$\displaystyle\to\frac{\lambda}{3!}\,H\,\int^{\texttt{t}}{\rm
d}\texttt{t}_{1}\int{\rm
d}^{3}x_{1}\,[a(\texttt{t}_{1})H]^{-2\alpha}\varphi_{+}^{3}\big{(}{\vec{x}}_{1}\big{)}$
$\displaystyle\hskip 40.0pt\
i\bigg{(}\Big{[}\varphi_{-}\big{(}{\vec{x}}_{1}\big{)},\varphi_{+}\big{(}{\vec{x}}\big{)}\Big{]}+\frac{[a(\texttt{t}_{1})H]^{3}}{[a(\texttt{t})H]^{3}}\Big{[}\varphi_{+}\big{(}{\vec{x}}_{1}\big{)},\varphi_{-}\big{(}{\vec{x}}\big{)}\Big{]}\bigg{)}$
$\displaystyle=\frac{\lambda}{3!}\frac{H}{3}\bigg{(}-\bigg{(}\\!-\frac{1}{2\alpha}+\log[a(\texttt{t})H]\bigg{)}+\frac{1}{3}\bigg{)}\varphi_{+}^{3}\big{(}{\vec{x}}\big{)}\,,$
(4.3)

where in the last line we expanded in $\alpha\ll 1$. Note that the scaling
dimension of $\varphi_{+}$ is still $\alpha$ and thus will provide the
necessary distance scale to make the log dimensionless inside of a correlator,
as is familiar from conventional dim reg.

Now we turn to exploring the same effect within the EFT direction by
calculating the time evolution of $\varphi_{+}$ using SdSET. Using Eq. 3.11
with $Q=\varphi_{+}$ and ${\cal H}_{\rm
int}=c_{3,1}\varphi_{+}^{3}\varphi_{-}/3!$ we have

$\displaystyle[\varphi_{+}]_{\lambda}({\vec{x}},\texttt{t})$
$\displaystyle=\frac{c_{3,1}}{3!}\int^{\texttt{t}}{\rm
d}\texttt{t}_{1}\int{\rm
d}^{3}x_{1}[aH(\texttt{t}_{1})]^{-2\alpha}\varphi_{+}^{3}({\vec{x}}_{1})i\Big{[}\varphi_{-}\big{(}{\vec{x}}_{1}\big{)},\varphi_{+}\big{(}{\vec{x}}\big{)}\Big{]}$
$\displaystyle=-\frac{c_{3,1}}{3!}\frac{1}{3}\bigg{(}\\!-\frac{1}{2\alpha}+\log[a(\texttt{t})H]\bigg{)}\varphi_{+}^{3}({\vec{x}})\,,$
(4.4)

where it is trivial to match the tree-level UV interaction to the EFT
interaction, such that

$\displaystyle c_{4,0}=c_{3,1}=\,...\,=\lambda+O\big{(}\lambda^{2}\big{)}\,.$
(4.5)

The equality between $c_{n,0}$ and $c_{n-m,m}$ found in matching is also fixed
by the reparametrization invariance of SdSET. We see that the EFT is capturing
the first term in the Taylor expansion of the UV theory given in Eq. 4.3, but
not the second.

The origin of the missing term is two-fold. First, we are only considering
correlations of $\varphi_{+}$ instead of the full UV field, which also
includes $\varphi_{-}$. This alone would not matter, since $\varphi_{-}$ is
suppressed by $[aH]^{-3}$. However, in order to organize the interactions
within the EFT, we removed the $c_{4,0}\varphi_{+}^{4}/4!$ term by a field
redefinition. Specifically, to remove the $c_{n,0}\varphi_{+}^{n}/n!$
operator, we take

$\varphi_{-}\to\varphi_{-}+\frac{c_{n,0}}{9(n-1)!}[aH]^{3-(n-1)\alpha}\varphi_{+}^{n-1}\
.$ (4.6)

Therefore, keeping track of the field redefinition implies that we should use

$\phi_{\rm
EFT}\equiv{\underline{\varphi}}=H\left([aH]^{-\alpha}\varphi_{+}+[aH]^{-\beta}\varphi_{-}+\frac{c_{4,0}}{9}\frac{1}{3!}[aH]^{-3\alpha}\varphi_{+}^{3}+\,...\,\right)\,,$
(4.7)

with $\alpha\to 0$ and $\beta\to 3$. Now the quantities on the RHS live purely
in the EFT. As a result $\varphi_{-}$ will not contribute to correlation
functions of ${\underline{\varphi}}$ because they are suppressed by powers of
$[aH]^{-3}$. Combining this with Eq. 4.4, we find

$\big{[}\,{\underline{\varphi}}\,\big{]}_{\lambda}=\varphi_{+}+\frac{1}{3!}\frac{H}{3}\bigg{(}c_{3,1}\bigg{(}\\!\frac{1}{2\alpha}-\log[aH]\bigg{)}+\frac{c_{4,0}}{3}\bigg{)}\varphi_{+}^{3}({\vec{x}})\,,$
(4.8)

where we dropped terms suppressed by powers of $[aH]$. Now we see that this
matches the UV expression in Eq. 4.3 when we use the tree-level matching
relations $c_{3,1}=c_{4,0}=\lambda$ given in Eq. 4.5.

This result also provides the map between SdSET and Refs. [31, 35], which
derived the soft behavior by explicitly expanding the UV in-in correlator in
the superhorizon limit. The tree-like structure they observe is a consequence
of our power counting as only $c_{n,1}$ is marginal and hence interactions
only include a single factor of $\varphi_{-}$. The nested set of commutators
in Eq. 3.4 ensures that the marginal operators always have a tree-like
structure. In SdSET, this is manifest from dynamical RG, and all the
additional finite terms arise from the field redefinitions.

By similar considerations, we must apply the field redefinition to match the
UV potential onto $V(\varphi_{+},\varphi_{-})$ in the EFT. Although
$\lambda\varphi_{+}^{4}$ has been removed, this procedure introduces higher
order terms, such as

$\displaystyle\frac{c_{3,1}}{3!}\varphi_{+}^{3}\varphi_{-}$
$\displaystyle\,\,\to\,\,\frac{c_{3,1}\,c_{4,0}}{9(3!)^{2}}[aH]^{3}\varphi_{+}^{6}$
(4.9a) $\displaystyle\frac{c_{2,2}}{4}[aH]^{-3}\varphi_{+}^{2}\varphi_{-}^{2}$
$\displaystyle\,\,\to\,\,\frac{c_{2,2}\,c_{4,0}}{18(3!)}\varphi_{+}^{5}\varphi_{-}\,.$
(4.9b)

Removing the first term will introduce a $\varphi_{+}^{7}\varphi_{-}$
interaction at order $\lambda^{3}$ and so on. As a result, our field
redefinition requires that

$V(\varphi_{+},\varphi_{-})\supset\frac{1}{3!}\varphi_{-}\left(c_{3,1}\varphi_{+}^{3}+\frac{c_{2,2}c_{4,0}}{18}\varphi_{+}^{5}+\frac{c_{2,2}c_{3,1}c_{4,0}}{162}\varphi_{+}^{7}+\,...\,\right)\
.$ (4.10)

Finally, using $c_{2,2}=c_{3,1}=c_{4,0}=\lambda$ from matching, we arrive at

$V(\varphi_{+},\varphi_{-})\supset\frac{\lambda}{3!}\varphi_{-}\left(\varphi_{+}^{3}+\frac{\lambda}{18}\varphi_{+}^{5}+\frac{\lambda^{2}}{162}\varphi_{+}^{7}+\,...\,\right)\,.$
(4.11)

or $c_{5,1}=\frac{\lambda^{2}}{18}\frac{5!}{3!}$ and
$c_{7,1}=\frac{\lambda^{3}}{162}\frac{7!}{(3!)}$.

The correction to $c_{5,1}$ is equivalent to the NLO corrections to the
effective potential calculated in Refs. [30] and [34] using complementary
techniques. While these two references approach this problem from different
perspectives, the wavefunction of the universe and the dS static patch
respectively, both effectively integrate out the decaying mode $\varphi_{-}$
which leads to an additional term in the potential. Instead, when
$\varphi_{-}$ is included, our corrections arise from insuring $\varphi_{-}$
does not mix with $\varphi_{+}$ at higher orders in perturbation theory. As
the dimensions of $\varphi_{+}$ and $\varphi_{-}$ are well separate for
$\alpha\to 0$, removing mixing can always be achieved by such a field
redefinition. Additionally, it is easy to determine the correction to
$c_{2n+1,1}\propto\lambda^{n}$ by repeated application of Eq. 4.6. Most
importantly, we do not integrate out $\varphi_{-}$ to ensure we have a local
action, rather than an open EFT for the growing mode alone [29].

What we have accomplished thus far is to determine the correct basis of
operators to match the EFT and UV descriptions. We have ensured that the
superhorizon limit of $\phi$ and ${\underline{\varphi}}$ agree as operators at
higher orders in $\lambda$. However, in order to match the correlators of the
UV theory, which include the subhorizon evolution, we will need to determine
the stochastic initial conditions beyond the Gaussian limit.

In order to correctly match the four-point function, we write the EFT
trispectrum to order $\lambda$ as444For convenience, we take $C_{\alpha}\to 1$
for the light fields and have dropped the additional constants that arise from
expanding $C_{\alpha}=1+\alpha(\gamma_{E}-2+\log
2)+{O}\big{(}\alpha^{2}\big{)}$. While this choice has no impact on the
physics (since the correct constants will appear in the initial conditions by
matching), it will simplify the algebra significantly.

$\displaystyle\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}_{C}^{\prime}$
$\displaystyle=\Big{\langle}\big{[}\,{\underline{\varphi}}\,\big{]}_{\lambda}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}_{C}+{\rm
permutations}$ $\displaystyle\hskip
14.0pt+\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}_{{\rm
IC}^{(1)}}$
$\displaystyle=\frac{\lambda\,H^{4}}{8(k_{1}k_{2}k_{3}k_{4})^{3}}\sum_{i}\frac{k_{i}^{3}}{3}\left(c_{3,1}\bigg{(}\\!\frac{1}{2\alpha}+\log\frac{k_{i}}{[aH]}\bigg{)}+\frac{c_{4,0}}{3}\right)$
$\displaystyle\hskip
14.0pt+\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}_{{\rm
IC}^{(1)}}\,,$ (4.12)

where $c_{3,1}=c_{4,0}=\lambda$ as before, and the subscript “$C$” denotes
that this is only the connected contributions. Matching this EFT expression to
the UV result in Eq. 3.9 fixes the non-Gaussian contribution to the initial
conditions:

$\displaystyle\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}_{{\rm
IC}^{(1)}}$
$\displaystyle=\frac{\lambda\,H^{4}}{8(k_{1}k_{2}k_{3}k_{4})^{3}}\Bigg{[}\frac{\sum_{i}k_{i}^{3}\big{(}-\frac{1}{2\alpha}+\gamma_{E}-2+\log\frac{k_{t}}{k_{i}}\big{)}}{3}$
$\displaystyle-\frac{k_{1}k_{2}k_{3}k_{4}}{k_{t}}$
$\displaystyle-\frac{1}{9}k_{t}^{3}+2\sum_{h<i<j}k_{h}k_{i}k_{j}+\frac{1}{3}k_{t}\Big{(}\sum
k_{i}^{2}-\sum_{i<j}k_{i}k_{j}\Big{)}\Bigg{]}\,,$ (4.13)

where $k_{t}=k_{1}+k_{2}+k_{3}+k_{4}$. Most significantly, all the time-
dependence of the full UV trispectrum is already captured by the EFT and, as
expected, the initial conditions are only required for matching the time-
independence contributions. This result is extended to the six-point function
in App. A as expected from general arguments.

Matching Derivative Operators: Before moving on the loop-level matching, let
us briefly comment on the case where the UV theory is itself an effective
theory. Specifically, we are only considering the case of a $\lambda\phi^{4}$
interaction in the UV, while in principle there could be a variety of higher
derivative (irrelevant) interactions as well. The first such operator we can
write down is $(\nabla_{\mu}\phi\nabla^{\mu}\phi)^{2}/M^{4}$. The full tree
level trispectrum was calculated in Ref. [56] and is given by

$\displaystyle\Big{\langle}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}_{\nabla^{4}}$
$\displaystyle=(2\pi)^{3}\delta\left(\sum_{i}\vec{k}_{i}\right)\frac{1}{M^{4}}\frac{H^{8}}{\prod_{i}2k_{i}^{3}}$
$\displaystyle\hskip
18.0pt\times\Bigg{[}-\frac{144k_{1}^{2}k_{2}^{2}k_{3}^{2}k_{4}^{2}}{k_{t}^{5}}-4\Bigg{(}\frac{12k_{1}k_{2}k_{3}k_{4}}{k_{t}^{5}}+\frac{3\prod_{i<j<l}k_{i}k_{j}k_{l}}{k_{t}^{4}}$
$\displaystyle\hskip
40.0pt+\frac{\prod_{i<j}k_{i}k_{j}}{k_{t}^{3}}+\frac{1}{k_{t}}\Bigg{)}\left(\left(\vec{k}_{1}\cdot\vec{k}_{2}\right)\left(\vec{k}_{3}\cdot\vec{k}_{4}\right)+2\text{
perms}\right)$ $\displaystyle\hskip
40.0pt+\left(\vec{k}_{1}\cdot\vec{k}_{2}\right)\left(\frac{4k_{3}^{2}k_{4}^{2}}{k_{t}^{3}}+\frac{12\left(k_{1}+k_{2}\right)k_{3}^{2}k_{4}^{2}}{k_{t}^{4}}+\frac{48k_{1}k_{2}k_{3}^{2}k_{4}^{2}}{k_{t}^{5}}\right)$
$\displaystyle\hskip 40.0pt+\text{5 perms}\Bigg{]}\,.$ (4.14)

What we notice right away is that this gives us a completely time-independent
result. It can therefore only be absorbed into the initial conditions of
$\varphi_{+}$:

$\displaystyle\Big{\langle}\varphi_{+}\big{(}\vec{k}_{1}\big{)}\varphi_{+}\big{(}\vec{k}_{2}\big{)}\varphi_{+}\big{(}\vec{k}_{3}\big{)}\varphi_{+}\big{(}\vec{k}_{4}\big{)}\Big{\rangle}_{\rm
IC}$ $\displaystyle\supset$ $\displaystyle
H^{-4}\Big{\langle}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\phi\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}_{\nabla^{4}}\,.$
(4.15)

This result illustrates a broader feature of physics in dS: higher derivatives
decouple at long wavelength and thus contribute, at most, time-independent
correlation functions. This property is made manifest in SdSET and must hold
in matching. As a result, a variety of possible UV theories only impact the
initial conditions and not the long wavelength dynamics. In this sense, our
choice to focus $\lambda\phi^{4}$ is not missing more complicated superhorizon
evolution. Instead, higher derivative terms are trivially matched in SdSET and
thus do not further illuminate the structure of the EFT.

###  4.2 One-loop Matching

When calculating loops in the UV theory, rather than the EFT, our EFT
regulator (dynamical dim reg) is not effective.555In a forthcoming paper [57],
it will be shown that when these integrals are transformed to Mellin space,
then it is indeed possible to implement dynamical dim reg in the full theory.
Loop calculations in the UV are not scaleless and therefore we will need to
regulate them differently than the EFT approach. This difference will be
absorbed into the matching calculating. In App. B, we match using a hard
cutoff in both theories for an example with consistent regulators in both and
find identical results, up to scheme dependent coefficients. To avoid the
usual challenges of working with a hard cutoff, we will use dimensional
regularization in the UV theory via

$\Big{\langle}\phi\big{(}{\vec{\scaleto{k}{7pt}}},\tau\big{)}\,\phi\big{(}\\!-\\!{\vec{\scaleto{k}{7pt}}},\tau^{\prime}\big{)}\Big{\rangle}=\frac{\pi}{4}H^{d-1}(-\tau)^{\frac{d}{2}}\left(-\tau^{\prime}\right)^{\frac{d}{2}}\text{H}_{\nu}\big{(}\\!-\\!k\tau\big{)}\text{H}_{\nu}^{\star}\big{(}\\!-\\!k\tau^{\prime}\big{)}\,,$
(4.16)

where $\nu=\sqrt{d^{2}/4-m^{2}/H^{2}}$. To simplify calculations, we can fix
$\nu=3/2$ for any dimension, and then can regulate integrals that appear via
an analytic continuation in $d$. We note that while this will regulate the
one-loop divergences that appear in this section, dim reg alone is not
sufficient in general, as we will see below.

We will begin with the one-loop power spectrum of the growing mode,
illustrated in the left side of Fig. 2. In the UV theory, a standard in-in
calculations gives us

$\displaystyle\Big{\langle}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}\,\big{)}\phi\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip 0.8pt\prime}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{(1)}$
$\displaystyle=\frac{\lambda}{4k^{3}}\frac{H^{2}}{3}\left(\frac{1}{\epsilon}+\log\frac{2k}{[aH]}+\gamma_{E}-2\right)\bigg{[}[aH]^{-\epsilon}\int\frac{{\rm
d}^{d}p}{(2\pi)^{d}}\frac{1}{2p^{3}}\bigg{]}\,.$ (4.17)

Regulating the IR by substituting $p^{2}\to p^{2}+K^{2}$ in the denominator,
we get

$\displaystyle\Big{\langle}\phi\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}\,\big{)}\phi\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip 0.8pt\prime}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{(1)}$
$\displaystyle=\frac{\lambda}{8\pi^{2}k^{3}}\frac{H^{2}}{3}\bigg{(}\frac{1}{\epsilon}+\log\frac{2k}{[aH]}+\gamma_{E}-2\bigg{)}\bigg{[}\frac{1}{\epsilon}-\log\frac{K}{[aH]}+\frac{1}{2}\big{(}\log
4\pi-\gamma_{E}\big{)}\bigg{]}\,.$

This is the complete one-loop power spectrum of the UV theory.

${\vec{\scaleto{k}{7pt}}}$${\vec{\scaleto{k}{7pt}}}\,^{\prime}$$t_{1}$${\vec{\scaleto{k}{7pt}}}_{1}$${\vec{\scaleto{k}{7pt}}}_{2}$${\vec{\scaleto{k}{7pt}}}_{3}$${\vec{\scaleto{k}{7pt}}}_{4}$$t_{1}$$t_{2}$time
Figure 2: Diagrams for the one-loop matching, as computed in the UV theory.
The horizontal line indicates a surface on constant conformal time $\tau_{0}$
on which our in-in correlators are evaluated. Left: One-loop power spectrum.
Right: One-loop trispectrum.

In SdSET, the one-loop power spectrum is given by several terms

$\displaystyle\Big{\langle}{\underline{\varphi}}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}\,\big{)}{\underline{\varphi}}\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip 0.8pt\prime}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{(1)}$
$\displaystyle=H^{2}[aH]^{-2\alpha}\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}\,\big{)}\varphi_{+}\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip 0.8pt\prime}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{(1)}+H^{2}[aH]^{-2\alpha}\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}\,\big{)}\varphi_{+}\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip 0.8pt\prime}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{\delta\alpha^{(1)}}$ $\displaystyle\hskip
13.0pt+2\frac{c_{4,0}}{9}\frac{1}{3!}H^{2}[aH]^{-4\alpha}\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}\,\big{)}\varphi_{+}^{3}\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip 0.8pt\prime}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{(0)}+\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}\big{)}{\underline{\varphi}}\big{(}{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\Big{\rangle}^{\prime}_{{\rm IC}^{(1)}}\,,$ (4.19)

where $c_{4,0}\sim\lambda$, and ${\underline{\varphi}}$ is given in Eq. 4.7.
The subscript $(n)$ labels the order in $\lambda$ in which correlator is
calculated and $\delta\alpha^{(m)}$ is the contribution to the correlator from
a shift in the value of $\alpha$ at order $\lambda^{m}$. The first term is the
one-loop power spectrum in the EFT,

$\displaystyle[aH]^{-2\alpha}\Big{\langle}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\big{)}\varphi_{+}\big{(}{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\Big{\rangle}^{\prime}_{(1)}$
$\displaystyle=-[aH]^{-4\alpha}\frac{1}{2\nu}\bigg{(}-\frac{1}{2\alpha}\bigg{)}\frac{\lambda}{4k^{3-2\alpha}}\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\frac{1}{p^{3-2\alpha}}$
$\displaystyle\to-\frac{1}{6\pi^{2}}\frac{\lambda}{4k^{3}}\bigg{(}-\frac{1}{2\alpha}-\log\frac{k}{[aH]}\bigg{)}\bigg{(}-\frac{1}{2\alpha}-\log\frac{K}{[aH]}\bigg{)}\,.$

We must also allow for the possibility that, due to matching, we will need to
correct the value of $\alpha$ from its free-field value. Substituting
$\alpha\to\alpha+\delta\alpha^{(1)}$ in Eq. 3.18a and expanding to linear
order, we have

$H^{2}[aH]^{-2\alpha}\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}\,\big{)}\varphi_{+}\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip 0.8pt\prime}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{\delta\alpha^{(1)}}=\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}\big{)}{\underline{\varphi}}\big{(}{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\Big{\rangle}_{(0)}\bigg{(}1+2\delta\alpha^{(1)}\log\frac{k}{[aH]}\bigg{)}\,,$
(4.21)

where have absorbed the $[aH]^{-2\delta\alpha}$ into the definition of this
term to derive the dimensionless argument of the log. From the EFT point of
view, $\delta\alpha^{(1)}$ is an unknown constant of order $\lambda$ to be
fixed by matching. Finally, we have the contribution to the
${\underline{\varphi}}$ power spectrum from the field redefinition:

$\displaystyle
2\frac{c_{4,0}}{9}\frac{1}{3!}H^{2}[aH]^{-4\alpha}\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}\,\big{)}\varphi_{+}^{3}\big{(}\hskip
0.8pt{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip 0.8pt\prime}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{(0)}\to\frac{c_{4,0}}{9}\frac{1}{2k^{3}}\frac{1}{2\pi^{2}}\bigg{(}-\frac{1}{2\alpha}-\log\frac{K}{[aH]}+\log
2\bigg{)}\,.$ (4.22)

We emphasize that since the coefficient of $\varphi_{+}^{3}$ in the definition
of ${\underline{\varphi}}$ is fixed by matching the superhorizon six-point
function, there is no additional freedom within this term that can be used to
match the 1-loop power spectrum.

The final term
$\delta\big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}\big{)}{\underline{\varphi}}\big{(}{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\big{\rangle}^{\prime}_{\rm IC}$ is time independent and
is determined by matching the time-independent part of the UV calculation.
Combining these results and using $c_{4,0}\to\lambda$, we find

$\displaystyle
H^{2}\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}\big{)}{\underline{\varphi}}\big{(}{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\Big{\rangle}^{\prime}_{(1)}=$
$\displaystyle\frac{H^{2}}{6\pi^{2}}\frac{\lambda}{4k^{3}}\bigg{(}\frac{1}{-2\alpha}+\log\frac{[aH]}{k}+\frac{2}{3}\bigg{)}\bigg{(}\frac{1}{-2\alpha}+\log\frac{[aH]}{K}\bigg{)}$
$\displaystyle+\delta\alpha^{(1)}\frac{H^{2}}{k^{3}}\log\frac{k}{a}+\frac{\lambda}{9}\frac{1}{2k^{3}}\frac{1}{2\pi^{2}}\log
2+\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}\big{)}{\underline{\varphi}}\big{(}{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\Big{\rangle}^{\prime}_{{\rm IC}^{(1)}}\ .$ (4.23)

Comparing this result to LABEL:eq:UVTwoPt1LCorr, we see that the time-
dependence of the UV and EFT agree after matching the single log coefficient
with

$\delta\alpha^{(1)}=\frac{\lambda}{8\pi^{2}}\frac{1}{3}\bigg{(}\gamma_{E}-\frac{7}{3}+\log
2-\frac{1}{2}\big{(}\log 4\pi-\gamma_{E}\big{)}\bigg{)}\,.$ (4.24)

This is a non-trivial result as we only have a single free parameter to match
the full time-dependence on the UV result. All the time-independent
contributions can be matched by adjusting the initial conditions, IC(1), which
in this case is just a renormalization of the amplitude of the two point
function, namely a shift in $C_{\alpha}$.

#### One-loop Trispectrum

Now we move on to the one-loop trispectrum, as illustrated in the right side
of Fig. 2. For our purposes here, it suffices to simply match the UV divergent
terms which result in time dependent $\log[aH]$ factors.666We remind the
reader that the “UV divergences” that we are isolating here take their origins
from the IR divergences of the full theory that are resummed by Stochastic
Inflation. The expansions being done to simplify the integrals that appear on
the UV theory side of this matching calculation amount to decomposing the
correlator via the method of regions [58, 59]. This will result in a
correction to the $c_{3,1}$ Wilson coefficient in the EFT. While the initial
conditions also receive corrections from matching the trispectrum, these would
contribute to Stochastic Inflation beyond NNLO, and so we will not compute
them here.

For convenience, we break the UV calculation into two terms

$\displaystyle{\cal I}_{4}$
$\displaystyle=\Big{\langle}\phi^{(2)}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\phi^{(1)}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\phi^{(2)}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\phi^{(1)}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}+{\rm
permutations}$ (4.25a) $\displaystyle{\cal K}_{4}$
$\displaystyle=\Big{\langle}\phi^{(3)}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\phi^{(1)}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\phi^{(1)}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}\phi^{(1)}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}+{\rm
permutations}\,,$ (4.25b)

where

$\displaystyle\phi^{(2)}({\vec{\scaleto{k}{7pt}}},t)$
$\displaystyle=i\int^{t}{\rm d}t_{1}\Big{[}H_{\rm
int}(t_{1}),\phi\big{(}{\vec{\scaleto{k}{7pt}}},t\big{)}\Big{]}$ (4.26a)
$\displaystyle\phi^{(3)}({\vec{\scaleto{k}{7pt}}},t)$
$\displaystyle=-\bigg{[}\int^{t_{1}}{\rm d}t_{2}H_{\rm
int}(t_{2}),\bigg{[}\int^{t}{\rm d}t_{1}H_{\rm
int}(t_{1}),\phi({\vec{\scaleto{k}{7pt}}},t)\bigg{]}\bigg{]}\,,$ (4.26b)

where $H_{\rm int}$ is given in Eq. 3.5. In what follows, we will show that
${\cal I}_{4}$ is UV finite, while ${\cal K}_{4}$ is UV divergent. Then we
will match this divergent contribution between the UV theory and the EFT to
derive the one-loop correction to $c_{3,1}$.

We begin by evaluating ${\cal I}_{4}$. Expanding in the limit $p\gg k_{i}$, we
have

$\displaystyle{\cal I}_{4}=\frac{1}{k_{2}^{3}k_{4}^{3}}$
$\displaystyle\int^{\tau}\frac{{\rm d}\tau_{1}}{(-\tau_{1})^{4}}\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}G\big{(}{\vec{\scaleto{k}{7pt}}}_{1},\tau,\tau_{1}\big{)}\frac{(1-ip\tau_{1})^{2}}{p^{3}}e^{i2p\tau_{1}}$
$\displaystyle\times\int^{\tau}\frac{{\rm
d}\tau_{2}}{(-\tau_{2})^{4}}\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}G\big{(}{\vec{\scaleto{k}{7pt}}}_{3},\tau,\tau_{2}\big{)}\frac{(1+ip\tau_{2})^{2}}{p^{3}}e^{-i2p\tau_{2}}$
$\displaystyle+{\rm permutations}\,.$ (4.27)

The integrals over $\tau_{1}$ and $\tau_{2}$ are independent and we can
evaluate them as usual. Expanding in $k_{i}\tau_{i}\ll 1$, we have

$\displaystyle\int^{\tau}\frac{{\rm
d}\tau_{1}}{(-\tau_{1})^{4}}G\big{(}{\vec{\scaleto{k}{7pt}}}_{1},\tau,\tau_{1}\big{)}(1-ip\tau_{1})^{2}e^{i2p\tau_{1}}$
$\displaystyle\to-\frac{1}{3}\int^{\tau}\frac{{\rm
d}\tau_{1}}{(-\tau_{1})^{4}}\big{(}\tau^{3}-\tau_{1}^{3}\big{)}(1-ip\tau_{1})^{2}e^{i2p\tau_{1}}$
$\displaystyle=\frac{(-i)}{3}\left(\log p\tau-\frac{11}{12}\right)\,.$ (4.28)

Putting this together yields

${\cal I}_{4}\simeq\frac{1}{k_{2}^{3}k_{4}^{3}}\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\frac{1}{p^{6}}\left(\log
p\tau-\frac{11}{12}\right)^{2}\,.$ (4.29)

This integral will not lead to any UV divergences. Furthermore, this result is
exact in $p\tau$ since we have only expanded in $k/p$. Therefore all the
subleading terms will be more UV convergent, and correspond to higher power
corrections in the EFT. This shows that all we need to include when matching
$c_{3,1}$ is the correction due to ${\cal K}_{4}$.

The second term, ${\cal K}_{4}$, gives rise to more interesting UV behavior.
Again expanding for large loop momentum $p\gg k_{i}$, we have

$\displaystyle{\cal K}_{4}$
$\displaystyle\simeq\frac{H^{4}}{32(k_{2}k_{3}k_{4})^{3}}\int\frac{{\rm
d}^{d}p}{(2\pi)^{d}}\int^{\tau}\frac{{\rm
d}\tau_{1}}{(-\tau_{1})^{d+1}}G\big{(}{\vec{\scaleto{k}{7pt}}}_{1},\tau,\tau_{1}\big{)}$
$\displaystyle\hskip 140.0pt\times 2{\rm Im}\int^{\tau_{1}}\frac{{\rm
d}\tau_{2}}{(-\tau_{2})^{4}}\frac{1}{p^{6}}(1+ip\tau_{1})^{2}(1-ip\tau_{2})^{2}e^{-i2p(\tau_{1}-\tau_{2})}$
$\displaystyle\hskip 14.0pt+{\rm permutations}\,.$ (4.30)

We have expressed the commutator acting on the loop momenta in $\phi^{(3)}$ in
terms of the imaginary part to simplify the calculation. In this form, the
time integrals can be evaluated exactly, giving

$\displaystyle{\cal K}_{4}$
$\displaystyle\simeq\frac{H^{4}\lambda^{2}}{16(k_{2}k_{3}k_{4})^{3}}\int\frac{{\rm
d}^{d}p}{(2\pi)^{d}}\frac{1}{p^{3}}\bigg{[}\frac{10}{81}-\frac{1}{27}\gamma_{E}(2+3\gamma_{E})-\frac{5}{36}\pi^{2}$
$\displaystyle\hskip
14.0pt+\frac{1}{9}\left(\log\frac{2p}{[aH]}\right)^{2}+(1+3\gamma_{E})\log\frac{2p}{[aH]}+\frac{4}{9}\log\frac{k}{[aH]}\bigg{]}+{\rm
permutations}$
$\displaystyle=\frac{H^{4}\lambda^{2}}{8(k_{2}k_{3}k_{4})^{3}}\frac{1}{4\pi^{2}}\left(\frac{2}{\epsilon}+\log\frac{k}{[aH]}+\log\frac{K^{2}}{4\pi}-\gamma_{E}\right)$
(4.31) $\displaystyle\hskip
85.35826pt\times\left[\frac{10}{81}-\frac{1}{27}\gamma_{E}(2+3\gamma_{E}+6\log
2)-\frac{5}{36}\pi^{2}+\frac{2\log 2+3\log^{2}2}{27}\right]$
$\displaystyle\hskip 142.26378pt+{O}\Big{(}(\log[aH])^{2}\Big{)}+{\rm
permutations}\,.$

In the last line, we are only writing the terms that are linear in $\log[aH]$
since this is what determines the contribution to the RG.

We can see the appearance of higher powers of log from the perturbative EFT
contribution to the one-loop trispectrum

$\displaystyle\Big{\langle}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}$
$\displaystyle\supset\frac{C_{\alpha}^{6}\sum_{i}k_{i}^{3}}{8(k_{1}k_{2}k_{3}k_{4})^{3-2\alpha}}\frac{1}{9}$
$\displaystyle\hskip 16.0pt\times\int^{\texttt{t}}{\rm
d}\texttt{t}_{1}\int^{\texttt{t}_{1}}{\rm
d}\texttt{t}_{2}\big{(}[a(\texttt{t}_{1})H][a(\texttt{t}_{2})H]\big{)}^{-2\alpha}\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\frac{C_{\alpha}^{2}}{2p^{3-2\alpha}}$
$\displaystyle\to\frac{\sum_{i}k_{i}^{3}}{8(k_{1}k_{2}k_{3}k_{4})^{3-2\alpha}}\frac{1}{9}\,\frac{1}{2}\left(-\frac{1}{2\alpha}+\log[aH]\right)^{2}$
$\displaystyle\hskip
16.0pt\times\frac{1}{2\pi^{2}}\left(-\frac{1}{2\alpha}-\log\frac{K}{[aH]}+\log
2\right)\,,$ (4.32)

where the compensating dimensionful factors that appear inside the $\log[aH]$
term come from expanding the prefactor in the small $\alpha$ limit. One can
check that this term matches the coefficient of the log3 divergence of the UV
calculation.

In order to match the linear log term, we need to keep track of the EFT field
redefinition to order $\lambda^{2}$. Specifically, we need

${\underline{\varphi}}=H\left([aH]^{-\alpha}\varphi_{+}+[aH]^{-\beta}\varphi_{-}+\frac{\lambda}{9}\frac{1}{3!}[aH]^{-3\alpha}\varphi_{+}^{3}+\frac{\lambda^{2}}{81}\frac{1}{3!}[aH]^{-5\alpha}\varphi_{+}^{5}\right)\
.$ (4.33)

to remove the $\varphi_{+}^{6}$ operator from the EFT potential. This
${O}\big{(}\lambda^{2}\big{)}$ term contributes to the trispectrum at one
loop:

$\displaystyle\Big{\langle}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{3}\big{)}{\underline{\varphi}}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}$
$\displaystyle\supset$
$\displaystyle\frac{\sum_{i}k_{i}^{3}}{(k_{1}k_{2}k_{3}k_{4})^{3}}\frac{\lambda^{2}}{81}\frac{5!}{3!2}\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\frac{1}{p^{3-2\alpha}}\,,$ (4.34)

which matches the leading UV term in Sec. 4.2, namely the log term
proportional to a factor of $10/81$. After matching this term, we see a fairly
complicated expression remains for the linear log. This can be absorbed into
the $c_{3,1}$ Wilson coefficient in the potential using Eq. 3.23, such that

$c_{3,1}\to\lambda-\frac{\lambda^{2}}{2\pi^{2}}\bigg{(}\frac{1}{9}\gamma_{E}(2+3\gamma_{E}+6\log
2)+\frac{5}{12}\pi^{2}-\frac{2\log 2+3\log^{2}2}{9}\bigg{)}\,.$ (4.35)

This is the one-loop matching correction to an EFT Wilson coefficient, in
analogy with Eq. 4.24 above. A priori, one might think that we have to
consider the divergence from the mixing of the operators $\varphi_{+}$ and
$\varphi_{+}^{3}$. In later sections, we will see that such a mixing is
equivalent to a shift in the potential of the form in Eq. 4.35, and therefore
these two interpretations of the logarithmic growth are related to each other
by a field redefinition.

###  4.3 Initial Conditions for Composite Operators

The above matching procedure is sufficient to regulate the correlation
function of $\phi$ and match $\varphi_{+}$ correlators at separated points.
Composite operators are defined when some of these operators are at coincident
points. In Fourier space, this involves a convolution integral which can
produce divergences that require renormalizing the composite operator itself.

Composite operators can be defined in this way purely within the EFT. Since we
are regulating loops in the EFT with dynamical dim reg, this implies that we
will need to know initial conditions for general $\alpha$. Fortunately, we
will be interested in limits where the momenta are hierarchical (UV
divergences of the momentum integrals), which simplifies the matching
considerably.

We will start by taking the tree-level initial conditions for the trispectrum,
and tying two of the legs together to form a $\varphi_{+}^{2}[{\vec{x}}=0]$
composite operator:

$\Big{\langle}\varphi_{+}^{2}[0]\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}\hskip
0.8pt{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\Big{\rangle}^{\prime}_{\rm
IC}=\int\frac{{\rm d}^{3}p}{(2\pi)^{3}}\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{p}}\,\big{)}\varphi_{+}\big{(}-{\vec{p}}-{\vec{\scaleto{k}{7pt}}}_{1}-{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\Big{\rangle}^{\prime}_{\rm
IC}\,,$ (4.36)

where we are interested in determining the integrand of the right hand side in
the limit $p\gg k_{i}$. For $\alpha=0$, we determined the initial conditions
exactly in Eq. 4.13. In order to regulate the UV divergence that comes from
tying the two legs together, we want to evaluate this for general $\alpha$,
while isolating the term of interest, which is proportional to
$P_{+}(k_{1})P_{+}(k_{2})$, where $P_{+}$ is defined by

$\Big{\langle}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\varphi_{+}\big{(}{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\Big{\rangle}=P_{+}(k)(2\pi)^{3}\delta\big{(}{\vec{\scaleto{k}{7pt}}}+{\\!\\!\vec{{\scaleto{\,\,k}{7pt}}^{\hskip
0.8pt\prime}}}\big{)}\,,$ (4.37)

so that $\big{(}P_{+}(k)\big{)}_{\text{tree}}=(2k^{3})^{-1}$. Since $k\ll p$,
the initial conditions will arise at the horizon crossing of the modes
carrying momentum $p$ when the $k_{i}$-modes are superhorizon. As discussed in
Sec. 4.2, we therefore match using massive mode functions for only the $k_{i}$
fields, where we can also Taylor expand in $k\tau\ll 1$. As a result, the
initial conditions are given by

$\displaystyle\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{p}}\,\big{)}\varphi_{+}\big{(}-{\vec{p}}-{\vec{\scaleto{k}{7pt}}}_{1}-{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\Big{\rangle}^{\prime}_{\text{IC}^{(1)}}\simeq\lambda
P_{+}(k_{1})P_{+}(k_{2})$ $\displaystyle\hskip 70.0pt\times\lim_{\tau_{0}\to
0}2\hskip 0.8pt{\rm Im}\int^{\tau_{0}}\frac{{\rm
d}\tau}{(-H\tau)^{4}}[aH]^{-2\alpha}\frac{1}{4p^{2}}(1-ip\tau)^{2}(1+ip\tau_{0})^{2}e^{i2p(\tau-\tau_{0})}\,,$
(4.38)

where $[aH]^{-2\alpha}=(-\tau)^{2\alpha}$. The RHS of this expression is the
calculation in the UV theory with the appropriate choice of masses for the
mode functions. We are implicitly evaluating the correlation function at a
time $\tau_{0}$ and extracting the $\tau_{0}$-independent piece in the
$\tau_{0}\to 0$ limit. Evaluating the integral, we find

$\displaystyle\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{p}}\,\big{)}\varphi_{+}\big{(}{\vec{p}}-{\vec{\scaleto{k}{7pt}}}_{1}-{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\Big{\rangle}^{\prime}_{\rm
IC^{(1)}}$ $\displaystyle\simeq$ $\displaystyle\lambda
P_{+}(k_{1})P_{+}(k_{2})$
$\displaystyle\frac{(1+2\alpha)\Gamma[-1+2\alpha]\sin\big{(}\frac{\pi}{2}(1-2\alpha)\big{)}}{2^{2\alpha}p^{3+2\alpha}}\
.$ (4.39)

Corrections to this result are suppressed by powers of $k_{i}/p$ which will
not contribute to the one-loop divergences that we will use to determine the
RG for composite operator mixing below in Sec. 5.1.

For two-loop divergences, one must determine the initial conditions (see Sec.
5.2 below):

$\Big{\langle}\varphi_{+}^{3}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{\rm IC^{(1)}}=\int\frac{{\rm
d}^{3}p_{1}}{(2\pi)^{3}}\int\frac{{\rm
d}^{3}p_{2}}{(2\pi)^{3}}\Big{\langle}\varphi_{+}{}_{\alpha_{1}}\big{(}{\vec{p}}_{1}\big{)}\varphi_{+}{}_{\alpha_{2}}\big{(}{\vec{p}}_{2}\big{)}\varphi_{+}{}_{\alpha_{3}}\big{(}-{\vec{\scaleto{k}{7pt}}}-{\vec{p}}_{1}-{\vec{p}}_{2}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{\rm IC^{(1)}}\,,\\\ $

where we will take the limit $p_{i}\gg k$, such that
$-\vec{k}-{\vec{p}}_{1}-{\vec{p}}_{2}\simeq-{\vec{p}}_{1}-{\vec{p}}_{2}$. We
calculate this contribution for general $\alpha_{i}$ by matching to the full
theory:

$\displaystyle\Big{\langle}\phi_{\alpha_{1}}\big{(}{\vec{p}}_{1}\big{)}\phi_{\alpha_{2}}\big{(}{\vec{p}}_{2}\big{)}\phi_{\alpha_{3}}\big{(}{\vec{p}}_{3}\big{)}\phi\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\Big{\rangle}^{\prime}_{(1)}$ $\displaystyle=2\hskip 0.8pt{\rm
Im}\bigg{[}u_{\nu_{1}}^{*}\big{(}{\vec{p}}_{1},\tau\big{)}u_{\nu_{2}}^{*}\big{(}{\vec{p}}_{2},\tau\big{)}u_{\nu_{3}}^{*}\big{(}{\vec{p}}_{3},\tau\big{)}u_{3/2}^{*}\big{(}{\vec{\scaleto{k}{7pt}}},\tau\big{)}$
$\displaystyle\hskip 14.0pt\times\int^{\tau}\frac{{\rm
d}\tau_{1}}{(-\tau_{1})^{4}}u_{\nu_{1}}\big{(}{\vec{p}}_{1},\tau_{1}\big{)}u_{\nu_{2}}\big{(}{\vec{p}}_{2},\tau_{1}\big{)}u_{\nu_{3}}\big{(}{\vec{p}}_{3},\tau_{1}\big{)}u_{3/2}\big{(}{\vec{\scaleto{k}{7pt}}},\tau_{1}\big{)}\bigg{]}\,,$

where

$u_{\nu}\big{(}{\vec{\scaleto{k}{7pt}}},\tau\big{)}=-i\hskip
0.8pte^{i\big{(}\nu+\frac{1}{2}\big{)}\frac{\pi}{2}}\frac{\sqrt{\pi}}{2}H(-\tau)^{3/2}\text{H}_{\nu}^{(1)}(-k\tau)\,,$
(4.41)

is the positive frequency mode for a field with $\alpha=\frac{3}{2}-\nu$.
Again, since the mode functions at horizon crossing behave as if they are
effectively massless, this contribution can be determine using massless mode
function of $p\tau_{1}$, but we must use the massive mode functions for
$p\tau$. The result of integrating over $\tau_{1}$ is

$\Big{\langle}\varphi_{+}\big{(}{\vec{p}}_{1}\big{)}\varphi_{+}\big{(}{\vec{p}}_{2}\big{)}\varphi_{+}\big{(}{\vec{p}}_{3}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\Big{\rangle}_{\rm
IC^{(1)}}^{\prime}=\lambda\,\frac{\prod_{i=1}^{3}C_{\alpha_{i}}\hskip
0.8ptp^{\alpha_{i}}}{12\big{(}p_{1}p_{2}p_{3}\big{)}^{3}}\bigg{(}\sum_{i}\kappa_{i}\hskip
0.8ptp_{i}^{3}-p_{1}p_{2}p_{3}+\sum_{i\neq j}p_{i}^{2}p_{j}\bigg{)}\,,$ (4.42)

where the correlators were matched at time $\tau$,

$\kappa_{i}=\frac{1}{3}\left(-\frac{1}{2\alpha}+\gamma_{E}-2+\log\frac{p_{t}}{p_{i}}\right)\
,$ (4.43)

and $p_{t}=p_{1}+p_{2}+p_{3}$.

$t_{1}$$t_{2}$${\vec{\scaleto{k}{7pt}}}_{1}$${\vec{\scaleto{k}{7pt}}}_{2}$${\vec{p}}$$-{\vec{p}}$${\vec{\scaleto{k}{7pt}}}_{3}$${\vec{\scaleto{k}{7pt}}}_{4}$time
Figure 3: Tree level pentaspectrum

Finally, we must also determined the tree-level six-point initial conditions,
illustrated in Fig. 3. We are specifically interested in the correlator

$\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{p}}\,\big{)}\varphi_{+}\big{(}-{\vec{p}}-{\vec{\scaleto{k}{7pt}}}_{t}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}\,,$
(4.44)

in the limit $p\gg k_{i}$, where
${\vec{\scaleto{k}{7pt}}}_{t}\equiv\sum_{i=1}^{4}{\vec{\scaleto{k}{7pt}}}_{i}$.
As in the case of the trispectrum, we will isolate the term

$\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{p}}\,\big{)}\varphi_{+}\big{(}-{\vec{p}}-{\vec{\scaleto{k}{7pt}}}_{t}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}_{\rm
IC^{(2)}}\supset\lambda^{2}\Gamma_{2,4}(p)P_{+}(k_{1})\,...\,P_{+}(k_{4})\,,$
(4.45)

such that

$\displaystyle\Gamma_{2,4}(p)$ $\displaystyle=\int^{\tau}\frac{{\rm
d}\tau_{1}}{(-\tau_{1})^{4-2\alpha}}\int^{\tau}\frac{{\rm
d}\tau_{2}}{(-\tau_{2})^{4-2\alpha}}\Big{\langle}\phi^{2}(p,\tau_{1})\phi^{2}(p,\tau)\phi^{2}(p,\tau_{2})\Big{\rangle}$
$\displaystyle\hskip 14.0pt-2\hskip 0.8pt{\rm Re}\int^{\tau}\frac{{\rm
d}\tau_{1}}{(-\tau_{1})^{4-2\alpha}}\int^{\tau_{1}}\frac{{\rm
d}\tau_{2}}{(-\tau_{2})^{4-2\alpha}}\Big{\langle}\phi^{2}(p,\tau)\phi^{2}(p,\tau_{1})\phi^{2}(p,\tau_{2})\Big{\rangle}\,.$
(4.46)

By direct calculation we find that

$\displaystyle\Gamma_{2,4}(p)=\frac{1}{216p^{3+4\alpha}}\bigg{[}$
$\displaystyle
16+4\gamma_{E}(-11+3\gamma_{E})+3\pi^{2}+4(-11+6\gamma_{E}+3\log 2)\log 2$
$\displaystyle+{O}\bigg{(}\log\frac{p}{[aH]}\bigg{)}\bigg{]}\,.$ (4.47)

##  5 Composite Operator Mixing

From our above discussion, we argued that corrections to Stochastic Inflation
are uniquely determined by the correlation functions of composite operators;
computing those that are relevant to correcting Stochastic Inflation up to
NNLO is the topic of this section. We will start by setting up the problem of
calculating composite operator renormalization. We are interested in operators
of the form $\varphi_{+}^{n}({\vec{x}})$. These are well-defined purely within
the EFT, so we must be able to discuss their correlation functions and
renormalizations given only the EFT data. Of course, the crucial information
is the initial conditions, which therefore depend on first matching to the UV.

In general, given the EFT field operator $\varphi_{+}({\vec{x}},\texttt{t})$,
we can always define a composite operator

$\varphi_{+}^{n}({\vec{x}})=\prod_{i=1}^{n}\int\frac{{\rm
d}^{3}p_{i}}{(2\pi)^{3}}e^{-i{\vec{p}}_{i}\cdot{\vec{x}}}\varphi_{+}({\vec{p}}_{i})\,.$
(5.1)

In a free theory, we find the structure:

$\displaystyle\Big{\langle}\varphi_{+}^{n}\big{(}{\vec{x}}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{n}\big{)}\Big{\rangle}$
$\displaystyle=n!\prod_{i=1}^{n}\int\frac{{\rm
d}^{3}p_{i}}{(2\pi)^{3}}e^{-i{\vec{p}}_{i}\cdot{\vec{x}}}(2\pi)^{3}\delta({\vec{\scaleto{k}{7pt}}}_{i}+{\vec{p}}_{i})P_{+}(k_{i})$
$\displaystyle=n!\hskip
0.8pte^{i\sum\vec{k}_{i}\cdot{\vec{x}}}P_{+}(k_{1})\,...\,P_{+}(k_{n})\,,$
(5.2)

where $P_{+}$ is defined in Eq. 4.37. Taking the Fourier transform,

$\varphi_{+}^{n}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip 0.8pt\big{)}=\int{\rm
d}^{3}x\,e^{i{\vec{\scaleto{k}{7pt}}}\cdot{\vec{x}}}\varphi_{+}^{n}\big{(}{\vec{x}}\big{)}\,,$
(5.3)

we have

$\Big{\langle}\varphi_{+}^{n}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{n}\big{)}\Big{\rangle}=n!\hskip
0.8ptP_{+}(k_{1})\,...\,P_{+}(k_{n})\hskip
0.8pt\delta\Big{(}{\vec{\scaleto{k}{7pt}}}+\sum{\vec{\scaleto{k}{7pt}}}_{i}\Big{)}\,.$
(5.4)

For simplicity, it will often be easiest to use
$\varphi_{+}^{n}({\vec{x}}=0)\equiv\varphi_{+}^{n}[0]$ to avoid the extra
$\delta$-function.

Given this definition, we see that we should be able to define all such
correlators of composite operators as simply integrals over the correlators of
$\varphi_{+}$. For example,

$\Big{\langle}\varphi_{+}^{n}\big{[}{\vec{x}}=0\big{]}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{m}\big{)}\Big{\rangle}=\prod_{i=1}^{n}\int\frac{{\rm
d}^{3}p_{i}}{(2\pi)^{3}}\Big{\langle}\varphi_{+}\big{(}{\vec{p}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{p}}_{n}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{m}\big{)}\Big{\rangle}\,.$
(5.5)

A very important feature of this formula is that the integrand on the RHS is
free of divergences, in the sense that we should have already regulated and
renormalized the expression. As a result, all of the renormalization of the
composite operator itself (and hence the matrix of anomalous dimensions) has
to be associated with integrals over ${\vec{p}}_{i}$ as opposed to the loop
integrals that appear in the calculation of
$\big{\langle}\varphi_{+}\big{(}{\vec{p}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{p}}_{n}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{m}\big{)}\big{\rangle}$
itself.

###  5.1 One-loop Corrections

#### Trispectrum

We will start by computing $b_{1}$, which is determined from the four-point
function via

$\displaystyle\Big{\langle}\varphi_{+}^{2}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\Big{\rangle}=\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{p}}\,\big{)}\varphi_{+}\big{(}-{\vec{p}}-{\vec{\scaleto{k}{7pt}}}_{1}-{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\Big{\rangle}^{\prime}\,.$
(5.6)

The relationship between this loop contribution and the tree-level trispectrum
is illustrated in Fig. 4. The contribution to the four-point function from the
time evolution already yields a $\log[aH]$. Therefore, any anomalous scaling
(which should also only be a single log) must arise from the initial
conditions. The relevant contribution was calculated in Eq. 4.39.

${\vec{p}}$$-{\vec{p}}$${\vec{\scaleto{k}{7pt}}}_{1}$${\vec{\scaleto{k}{7pt}}}_{2}$${\vec{p}}$$\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{1})$$\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{2})$$\varphi_{+}^{2}[0]$
time Figure 4: One loop correction to $\varphi_{+}^{2}$ that looks like an
anomalous dimension ($\varphi_{+}^{2}\to\varphi_{+}^{2}$). We start from the
tree level trispectrum and integrate over two of the fields to form the
composite operator $\varphi_{+}^{2}$. Left: The Witten diagram with a boundary
at future infinity. Right: The Feynman diagram with the same momentum flow.

Performing the integration over $p$ using dynamical dim reg, we find

$\displaystyle\Big{\langle}\varphi_{+}^{2}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\Big{\rangle}$
$\displaystyle=\lambda\hskip 0.8ptP_{+}(k_{1})P_{+}(k_{2})$
$\displaystyle\hskip
14.0pt\times\frac{(1+2\alpha)\Gamma[-1+2\alpha]\sin\big{(}\frac{\pi}{2}(1-2\alpha)\big{)}}{2^{2\alpha}}\frac{K^{-2\alpha}\pi^{3/2}\Gamma[\alpha]}{\Gamma\big{[}\frac{3}{2}+\alpha\big{]}}\,.$
(5.7)

Expanding for $\alpha\to 0$ we have

$\displaystyle\Big{\langle}\varphi_{+}^{2}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{2}\big{)}\Big{\rangle}$
$\displaystyle=\lambda\hskip
0.8ptP_{+}(k_{1})P_{+}(k_{2})\bigg{(}\frac{1}{48\pi^{2}\alpha^{2}}+\frac{(4-3\gamma_{E}-3\log
K)}{72\pi^{2}\alpha}+{\rm finite}\bigg{)}$
$\displaystyle\to\frac{\lambda}{36\pi^{2}}\hskip
0.8ptP_{+}(k_{1})P_{+}(k_{2})(4-3\gamma_{E})\bigg{(}\frac{1}{2\alpha}-\log\frac{[aH]}{k_{i}}\bigg{)}+\ldots\
.$ (5.8)

We can repeat this calculation with $\varphi_{+}^{n}[0]$ to determine the one-
loop anomalous dimension for all $n$. Keeping track of combinatorics, one
finds

$\displaystyle\Big{\langle}\varphi_{+}^{n}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{n}\big{)}\Big{\rangle}$
$\displaystyle\supset\frac{\lambda}{36\pi^{2}}\binom{n}{2}n!\hskip
0.8ptP_{+}(k_{1})\,...\,P_{+}(k_{n})$ $\displaystyle\hskip
14.0pt\times\sum_{i}(4-3\gamma_{E})\bigg{(}\frac{1}{2\alpha}-\log\frac{[aH]}{k_{i}}+{\cal
O}\left(\frac{1}{\alpha^{2}}\right)+{\rm finite}\bigg{)}\,.$ (5.9)

This NLO $\log[aH]$ dependence can be resummed by including the following
correction in the dynamical RG:

$b_{1}=-\frac{\lambda}{36\pi^{2}}(4-3\gamma_{E})\,.$ (5.10)

#### Six-point

Our next task is to compute $b_{2}$, which we determine by evaluating the six
point function,

$\displaystyle\Big{\langle}\varphi_{+}^{2}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}=\int\\!\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\bigg{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{p}}\,\big{)}\varphi_{+}\bigg{(}\\!\\!-{\vec{p}}-\sum_{i=1}^{4}\vec{k}_{i}\bigg{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\bigg{\rangle}^{\prime}\,,$
(5.11)

with $p\gg k_{i}$. The relationship between this one-loop contribution and the
tree-level six-point function is illustrated in Fig. 5. As before, only the
initial conditions arising from

$\Big{\langle}\varphi_{+}\big{(}\hskip
0.8pt{\vec{p}}\,\big{)}\varphi_{+}\big{(}-{\vec{p}}-{\vec{\scaleto{k}{7pt}}}_{t}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}^{\prime}\supset\lambda^{2}\hskip
0.8pt\Gamma_{2,4}(p)P_{+}(k_{1})\,...\,P_{+}(k_{4})\,,$ (5.12)

will contribute to the operator mixing, where $\Gamma_{2,4}(p)$ was calculated
in Eq. 4.47. Performing the momentum integration using dynamical dim reg, we
find

$\displaystyle\lambda^{2}\int\frac{{\rm d}^{3}p}{(2\pi)^{3}}\Gamma_{2,4}(p)$
$\displaystyle=\left(\frac{1}{8\pi^{2}\alpha}-\frac{1}{2\pi^{2}}+\,...\,\right)\left[b_{2,4}+{O}\left(\frac{1}{\alpha}\right)\right]+{\rm
finite}\,,$ (5.13)

where

$b_{2,4}=\frac{\lambda^{2}}{216}\Big{[}16+4\gamma_{E}(3\gamma_{E}-11)+3\pi^{2}+4\log
2(6\gamma_{E}+3\log 2-11)\Big{]}\ .$ (5.14)

Now we restore the factor $(k_{1}k_{2}k_{3}k_{4})^{\alpha}[aH]^{-4\alpha}$
associated with $P(k_{i})$ for general $\alpha$ and take the limit $\alpha\to
0$ to get

$\displaystyle\Big{\langle}\varphi_{+}^{2}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{4}\big{)}\Big{\rangle}$
$\displaystyle\supset
b_{2,4}P_{+}(k_{1})\,...\,P_{+}(k_{4})\left[\frac{1}{8\pi^{2}\alpha}+\frac{1}{2\pi^{2}}\sum_{i=1}^{4}\frac{1}{4}\log\frac{k_{i}}{[aH]}\right]\,.$
(5.15)

Repeating this calculation for $\varphi_{+}^{n}$ we have

$\displaystyle\Big{\langle}\varphi_{+}^{n}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{n+2}\big{)}\Big{\rangle}$
$\displaystyle\supset b_{2,4}\hskip 0.8pt\binom{n}{2}(n+2)!\hskip
0.8ptP_{+}(k_{1})\,...\,P_{+}(k_{n+2})$ $\displaystyle\hskip
22.0pt\times\left[\frac{1}{8\pi^{2}\alpha}+\frac{1}{2\pi^{2}}\sum_{i=1}^{4}\frac{1}{4}\log\frac{k_{i}}{[aH]}\right]\,.$
(5.16)

This NLO $\log[aH]$ dependence can be resummed by including the following
correction in the dynamical RG:

$b_{2}=\frac{1}{2\pi^{2}}\frac{\lambda^{2}}{216}\Big{[}16+4\gamma_{E}(3\gamma_{E}-11)+3\pi^{2}+4\log
2\,(6\gamma_{E}+3\log 2-11)\Big{]}\,.$ (5.17)

${\vec{\scaleto{k}{7pt}}}_{1}$${\vec{\scaleto{k}{7pt}}}_{2}$${\vec{p}}$$-{\vec{p}}$${\vec{\scaleto{k}{7pt}}}_{3}$${\vec{\scaleto{k}{7pt}}}_{4}$${\vec{p}}$$\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{1})$$\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{2})$$\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{3})$$\varphi_{+}({\vec{\scaleto{k}{7pt}}}_{4})$$\varphi_{+}^{2}[0]$
time Figure 5: Diagram of contribution one loop contribution to $\Gamma_{2,4}$
($\varphi_{+}^{2}\to\varphi_{+}^{4}$). We start from the tree level
pentaspectrum (6 points function) and integrate over two of the fields to form
the composite operator $\varphi_{+}^{2}$. Left: The Witten diagram with a
boundary at future infinity. Right: The Feynman diagram with the same momentum
flow.

###  5.2 Two-loop Corrections

Next, we move to the calculation of the two-loop anomalous dimension that
generates the NNLO non-Gaussian noise term for Stochastic Inflation. This
represents a novel contribution which is calculated here for the first time.
In particular, our goal is to calculate:

$\Big{\langle}\varphi_{+}^{3}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\Big{\rangle}=\int\frac{{\rm d}^{3}p_{1}\,{\rm d}^{3}p_{2}\,{\rm
d}^{3}p_{3}}{(2\pi)^{9}}\Big{\langle}\varphi_{+}\big{(}{\vec{p}}_{1}\big{)}\varphi_{+}\big{(}{\vec{p}}_{2}\big{)}\varphi_{+}\big{(}{\vec{p}}_{3}\big{)}\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\Big{\rangle}\,,$ (5.18)

where $p_{1,2,3}\gg k$. The relationship between this two-loop contribution
and the tree-level trispectrum is illustrated in Fig. 6. The correlation
function on the RHS was calculated in Eq. 4.42 for general $\alpha_{i}$ (for
each $p_{i}$) such that we can regulate the integral with dynamical dim reg.
Making the above substitution

$\displaystyle\Big{\langle}\varphi_{+}^{3}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\Big{\rangle}\supset P_{+}(k)\int\frac{{\rm d}^{3}p_{1}\,{\rm
d}^{3}p_{2}}{(2\pi)^{6}}$
$\displaystyle\lambda\frac{\prod_{i=1}^{3}C_{\alpha_{i}}p^{\alpha_{i}}}{12(p_{1}p_{2}p_{3})^{3}}\left(\kappa\sum_{i}p_{i}^{3}-p_{1}p_{2}p_{3}+\sum_{i\neq
j}p_{i}^{2}p_{j}\right)\,,$ (5.19)

where $\kappa$ is fixed by the full calculation for the reasons described
above, see Eq. 4.43. We note, that this result will require us to calculate
several integrals of the form

$I_{3}=\int\frac{{\rm d}^{3}p_{1}\,{\rm
d}^{3}p_{2}}{(2\pi)^{6}}\frac{1}{(p_{1}^{2})^{a}(p_{2}^{2})^{b}(({\vec{p}}_{1}+{\vec{p}}_{2})^{2})^{c}}\,,$
(5.20)

where $a$, $b$ and $c$ are half integers when $\alpha=0$. We can evaluate this
integral as follows:

$\displaystyle I_{3}$
$\displaystyle=\frac{\Gamma[b+c]}{\Gamma[b]\Gamma[c]}\int_{0}^{1}{\rm
d}x\int\frac{{\rm d}^{3}p_{1}}{(2\pi)^{3}}\frac{1}{p_{1}^{2a}}\int\frac{{\rm
d}^{3}\bar{p}_{2}}{(2\pi)^{3}}\frac{x^{b-1}(1-x)^{c-1}}{(\bar{p}_{2}^{2}+x(1-x)p_{1}^{2})^{b+c}}$
$\displaystyle=\frac{\Gamma[b+c-\tfrac{3}{2}]}{\Gamma[b]\Gamma[c]}\frac{1}{(4\pi)^{3/2}}\int\frac{{\rm
d}^{3}p_{1}}{(2\pi)^{3}}\frac{1}{p_{1}^{2a+2b+2c-3}}\int{\rm
d}x\frac{x^{b-1}(1-x)^{c-1}}{(x(1-x))^{b+c-3/2}}$
$\displaystyle=\frac{\Gamma[b+c-\tfrac{3}{2}]}{\Gamma[b]\Gamma[c]}\frac{\Gamma[\tfrac{3}{2}-b]\Gamma[\tfrac{3}{2}-c]}{\Gamma[3-b-c]}\frac{1}{(4\pi)^{3}}\frac{\Gamma[a+b+c-3]}{\Gamma[a+b+c-\tfrac{3}{2}]}K^{6-2a-2b-2c}\,,$
(5.21)

where we introduced an IR regulator $K$ as we did for our 1-loop divergence.
This IR regulator is only needed when $a+b+c\simeq 3$ (the integral vanishes
by dynamical dim reg otherwise). One should also notice that enforcing
$a+b+c\to 3$ restores the invariance under permutations of $a$, $b$ and $c$.

${\vec{p}}_{1}$${\vec{p}}_{2}$$-{\vec{p}}_{1}-{\vec{p}}_{2}$${\vec{\scaleto{k}{7pt}}}$$\varphi_{+}({\vec{\scaleto{k}{7pt}}})$$\varphi_{+}^{3}[0]$
${\vec{p}}_{1}$${\vec{p}}_{2}$$-{\vec{p}}_{1}-{\vec{p}}_{2}$time Figure 6:
Diagram of contribution two loop contribution to the mixing
$\varphi_{+}^{3}\to\varphi_{+}$. We start from the tree level trispectrum and
integrate over three of the fields to form the composite operator
$\varphi_{+}^{3}$. We first drew this as a Witten diagram with a boundary at
future infinity. At the bottom we have shown the Feynman diagram with the same
momentum flow.

We always have at least one log from $\Gamma[a+b+c-3]$. Therefore, the only
contributions that are not $\log^{2}$ (or higher) are those where all the
other $\Gamma$ functions are finite. Up to permutations, there are three
relevant cases ($i$) $a=3/2$, $b=3/2$, $c=0$, ($ii$) $a=3/2$, $b=1$, $c=1/2$,
and (iii) $a=b=c=1$. Only ($iii$) is a single log as expected from above.
Isolating just the single log term, for example using $a=1-\alpha/2$ with
$b=c=1$, we get

$\Big{\langle}\varphi_{+}^{3}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}\hskip
0.8pt\big{)}\Big{\rangle}_{(1)}=-P_{+}(k)\hskip
0.8pt\frac{\lambda}{12}\frac{1}{16\pi^{2}}\bigg{(}-\frac{1}{\alpha}+\log\frac{[aH]}{k}+\log\frac{k}{K}+\,...\,\bigg{)}+{O}\bigg{(}\frac{1}{\alpha^{2}}\bigg{)}\,.$
(5.22)

Repeating this calculation for $\varphi_{+}^{n}$ we have

$\displaystyle\Big{\langle}\varphi_{+}^{n}[0]\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{1}\big{)}\,...\,\varphi_{+}\big{(}{\vec{\scaleto{k}{7pt}}}_{n-2}\big{)}\Big{\rangle}_{(1)}$
$\displaystyle\supset-\frac{\lambda}{192\pi^{2}}\binom{n}{3}(n-2)!P_{+}(k_{1})\,...\,P_{+}(k_{n-2})$
$\displaystyle\hskip
70.0pt\times\sum_{i}\bigg{(}-\frac{1}{\alpha}-\sum_{i}\log\frac{k_{i}}{[aH]}\bigg{)}\,.$
(5.23)

This NNLO $\log[aH]$ dependence can be resummed by including the following
correction in the dynamical RG:

$d_{0}=\frac{\lambda}{192\pi^{2}}\,.$ (5.24)

###  5.3 Stochastic Inflation at NNLO

Now we have computed all the necessary pieces. To summarize our results, we
have found that at NNLO, the Fokker-Planck equation for Stochastic Inflation
becomes

$\displaystyle\frac{\partial}{\partial\texttt{t}}P(\varphi_{+},\texttt{t})$
$\displaystyle=\frac{1}{3}\frac{\partial}{\partial\varphi_{+}}\bigg{[}\partial_{\varphi_{-}}V(\varphi_{+},\varphi_{-})\Big{|}_{\varphi_{-}=0}P(\varphi_{+},\texttt{t})\bigg{]}$
$\displaystyle\hskip
13.0pt+\frac{1}{2}\frac{\partial^{2}}{\partial\varphi_{+}^{2}}\Big{[}(b_{0}+b_{1}\varphi_{+}^{2}+b_{2}\varphi_{+}^{4})P(\varphi_{+},\texttt{t})\Big{]}+\frac{1}{3!}\frac{\partial^{3}}{\partial\varphi_{+}^{3}}\Big{(}d_{0}\hskip
0.8pt\varphi_{+}P(\varphi_{+},\texttt{t})\Big{)}\,,$ (5.25)

where

$\displaystyle V(\varphi_{+},\varphi_{-})$
$\displaystyle=\frac{\lambda}{3!}\varphi_{-}\left(\varphi_{+}^{3}+\frac{\lambda}{18}\varphi_{+}^{5}+\frac{\lambda^{2}}{162}\varphi_{+}^{7}+\,...\,\right)$
(5.26a) $\displaystyle b_{0}$ $\displaystyle=\frac{1}{4\pi^{2}}$ (5.26b)
$\displaystyle b_{1}$
$\displaystyle=-\frac{\lambda}{36\pi^{2}}(4-3\gamma_{E})$ (5.26c)
$\displaystyle b_{2}$
$\displaystyle=\frac{1}{2\pi^{2}}\frac{\lambda^{2}}{216}\Big{[}16+4\gamma_{E}(3\gamma_{E}-11)+3\pi^{2}+4\log
2\,(6\gamma_{E}+3\log 2-11)\Big{]}$ (5.26d) $\displaystyle d_{0}$
$\displaystyle=\frac{\lambda}{192\pi^{2}}\,.$ (5.26e)

The rest of this section is devoted to expressing this result in a
particularly simple basis. As we have emphasized above, the coefficient
$b_{1}$ and $b_{2}$ are basis dependent, in the sense that they can be changed
by taking a field redefinition. For the purposes of solving for the NNLO
equilibrium distribution, we will find it useful to first perform a field
redefinition that moves their effects into the potential. For contrast, we
note that the coefficient $d_{0}$ is basis independent, and furthermore field
redefinitions of this $(\partial/\partial\varphi_{+})^{3}$ term will only
induce higher order (NNNLO) terms that we will neglect.

Concretely, we want to redefine the field
${\tilde{\varphi}_{+}}=f(\varphi_{+})$ such that
$\tilde{b}_{1}=\tilde{b}_{2}=0$. Using $P=({\rm d}{\tilde{\varphi}_{+}}/{\rm
d}\varphi_{+})\tilde{P}$ under such a field redefinition, we see that

$\displaystyle\frac{1}{2}\frac{\partial^{2}}{\partial\varphi_{+}^{2}}$
$\displaystyle\Big{[}\big{(}b_{0}+b_{1}\varphi_{+}^{2}+b_{2}\varphi_{+}^{4}\big{)}P(\varphi_{+},\texttt{t})\Big{]}\,\to\,$
$\displaystyle\frac{1}{2}\frac{\partial^{2}}{\partial{\tilde{\varphi}_{+}}^{2}}\Bigg{[}\bigg{(}\frac{{\rm
d}{\tilde{\varphi}_{+}}}{{\rm
d}\varphi_{+}}\bigg{)}^{2}\big{(}b_{0}+b_{1}\varphi_{+}^{2}+b_{2}\varphi_{+}^{4}\big{)}P({\tilde{\varphi}_{+}},\texttt{t})\Bigg{]}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\frac{\partial}{\partial{\tilde{\varphi}_{+}}}\Bigg{[}\frac{{\rm
d}^{2}{\tilde{\varphi}_{+}}}{{\rm d}\varphi_{+}^{2}}\frac{{\rm
d}{\tilde{\varphi}_{+}}}{{\rm
d}\varphi_{+}}\big{(}b_{0}+b_{1}\varphi_{+}^{2}+b_{2}\varphi_{+}^{4}\big{)}P({\tilde{\varphi}_{+}},\texttt{t})\Bigg{]}\,.$
(5.27)

Next, in order to set $\tilde{b}_{1}=\tilde{b}_{2}=0$, we define
${\tilde{\varphi}_{+}}$ so that

$\displaystyle\left(\frac{{\rm d}{\tilde{\varphi}_{+}}}{{\rm
d}\varphi_{+}}\right)^{2}\big{(}b_{0}+b_{1}\varphi_{+}^{2}+b_{2}\varphi_{+}^{4}\big{)}=b_{0}\,,$
(5.28)

We can integrate this equation to determine

${\tilde{\varphi}_{+}}=\varphi_{+}-\frac{b_{1}}{6b_{0}}\varphi_{+}^{3}+\frac{3b_{1}^{2}-4b_{0}b_{2}}{4b_{0}^{2}}\varphi_{+}^{5}\quad\to\quad\varphi_{+}\simeq{\tilde{\varphi}_{+}}+\frac{b_{1}}{6b_{0}}{\tilde{\varphi}_{+}}^{3}\,.$
(5.29)

The remaining term is then determined to be

$\displaystyle\frac{{\rm d}^{2}{\tilde{\varphi}_{+}}}{{\rm
d}\varphi_{+}^{2}}\frac{{\rm d}{\tilde{\varphi}_{+}}}{{\rm
d}\varphi_{+}}\big{(}b_{0}+b_{1}\varphi_{+}^{2}+b_{2}\varphi_{+}^{4}\big{)}$
$\displaystyle=-\frac{1}{2}\frac{b_{0}}{b_{0}+b_{1}\varphi_{+}^{2}+b_{2}\varphi_{+}^{4}}\big{(}2b_{1}\varphi_{+}+4b_{2}\varphi_{+}^{3}\big{)}$
$\displaystyle\simeq-