# Roles of boundary and equation-of-motion terms in cosmological correlation
functions

Ryodai<EMAIL_ADDRESS>Shinji
<EMAIL_ADDRESS>Yusuke<EMAIL_ADDRESS>1
Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo
169-8555, Japan  2 Waseda Institute for Advanced Study, Waseda University,
1-21-1 Nishi Waseda, Shinjuku, Tokyo 169-0051, Japan

###### Abstract

We revisit the properties of total time-derivative terms as well as terms
proportional to the free equation of motion (EOM) in a Schwinger-Keldysh
formalism. They are relevant to the correct calculation of correlation
functions of curvature perturbations in the context of inflationary Universe.
We show that these two contributions to the action play different roles in the
operator or the path-integral formalism, but they give the same correlation
functions as each other. As a concrete example, we confirm that the
Maldacena’s consistency relations for the three-point correlation function in
the slow-roll inflationary scenario driven by a minimally coupled canonical
scalar field hold in both the operator and path-integral formalisms. We also
give some comments on loop calculations.

††preprint: WUCG-24-03

##  I Introduction

The cosmological perturbation theory has been developed as a tool to study the
prediction e.g., of inflation models in the early Universe [1, 2, 3]. The
spectra of primordial perturbations generated during inflation can be tested
via cosmological observations such as cosmic microwave background temperature
anisotropies [4, 5]. In inflationary Universe, the quantum origin of those
perturbations should be addressed by quantum field theory (QFT) on a quasi de-
Sitter background (see e.g., Ref. [6]). One of the differences from the
S-matrix theory for the scattering of particles in Minkowski spacetime is the
use of a Schwinger-Keldysh (or the in-in or the expectation value) formalism,
as we evaluate the expectation values of operators rather than transition
amplitudes.

In the Schwinger-Keldysh formalism, there is a notion of the final spacelike
surface at which we often insert operators that we evaluate. The presence of
such a final surface does not allow us to freely neglect total time-derivative
terms in the Lagrangian since they appear in general as boundary operators.
Furthermore, in inflationary models, it is often convenient to use
integration-by-part techniques to reduce the action such that the interaction
vertices are proportional to powers of small slow-roll parameters, which makes
clear the validity of the perturbative approach. Such a procedure leaves the
aforementioned total time-derivative terms as well as terms that are
proportional to the free equation of motion (EOM) of fields in the Lagrangian,
which we call the EOM terms. In the first result of QFT calculations of the
three-point correlation function of cosmological perturbations performed by
Maldacena [7], he used a field redefinition technique to remove both the
boundary and the EOM terms simultaneously.

In this paper, we will revisit the question of how the boundary and EOM terms
play their roles in the practical calculations of the cosmological correlation
functions. It has been known that, even without performing the field
redefinition, the boundary terms give the same result as the field
redefinition while it is often claimed that the EOM terms play no roles [8, 9,
10, 11]. However, as we will show below, such a claim is correct only within
the operator formalism but not within the path-integral formalism. In the
path-integral formalism, the boundary terms do not contribute to the
correlation function by setting the final slice time $\tau_{f}$ later than
$\tau_{*}$ at which the expectation value of operators is evaluated, but the
EOM terms instead become nonnegligible, with which we find the same result as
the operator formalism.

One of the reasons why we address such a rather technical issue is the recent
discussion regarding the presence or absence of large one-loop corrections to
the scalar perturbation in inflationary models with a short ultra-slow-roll
phase between two slow-roll regimes [12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30]. The core of the argument of Ref. [12] is
that there is a possibly large coupling
$\eta^{\prime}\zeta^{\prime}\zeta^{2}/2$ in the cubic-order action, where
$\eta$ is one of the slow-roll parameters and a prime denotes the derivative
with respect to conformal time. This can generate a large one-loop
contribution to the two-point correlation function around the time of phase
transitions at which $|\eta^{\prime}|$ becomes very large.

On the other hand, in Refs. [25, 27], it was argued that including
contributions from some total time-derivative terms cancels seemingly large
contributions from the above coupling. Furthermore, it was shown that
Maldacena’s consistency relation can be reproduced only when the total time-
derivative terms are taken into account [20, 27]. In light of such
observations, it would be worthwhile to have a better understanding of the
roles of boundary and EOM terms for the computation of higher-order
corrections. Note also that the coupling
$\eta^{\prime}\zeta^{\prime}\zeta^{2}/2$ in the bulk action is a consequence
of the repeated applications of integration-by-part techniques, which yields
multiple total time-derivative terms. Hence, one expects that the bulk and the
“boundary terms” are not independent of each other.

The rest of the paper is organized as follows. In Sec. II, we discuss how the
boundary terms and the EOM terms contribute to the correlation functions
within the operator and the path-integral formalisms, respectively. In Sec.
III, we then illustrate the roles of those terms by computing three-point
correlation functions and show that they may play important roles in
reproducing consistency relations for various correlation functions. We also
give a few comments on higher-order corrections in Sec. IV. Sec. V is devoted
to conclusions. In appendix A, we note an identity with which the bulk,
boundary, and EOM terms can be changed. Appendix B is almost independent of
the other part, and we discuss some issues of total time-derivative terms in
the Hamiltonian, in general.

##  II Roles of boundary and EOM terms in Schwinger-Keldysh formalism

In this paper, we are interested in the correlation functions of scalar
perturbations on a spatially-flat Friedmann-Lemaître-Robertson-Walker (FLRW)
background given by the line element
$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}$, where $a(t)$ is the scale
factor that depends on the cosmic time $t$. In particular, we consider a
canonical scalar field $\phi$ minimally coupled to gravity, which is described
by the action

$S=\int d^{4}x\sqrt{-g}\left[\frac{M_{\rm
Pl}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]\,,$
(1)

where $M_{\rm Pl}$ is the reduced Planck mass, $R$ is the Ricci scalar, and
$V(\phi)$ is a scalar potential. The scalar field has a perturbation
$\delta\phi(t,x^{i})$ on the time-dependent background value $\phi(t)$. For
the perturbed line element, we choose the Arnowitt-Deser-Misner metric
expressed in the form

$ds^{2}=-N^{2}dt^{2}+a^{2}(t)e^{2\zeta(t,x^{i})}\delta_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right)\,,$
(2)

where $N$ is the lapse, $N^{i}$ is the shift, and $\zeta$ is the curvature
perturbation. We choose the unitary gauge $\delta\phi=0$ and integrate out $N$
and $N^{i}$ from the action by using their constraint equations of motion.

Expanding the action up to the second order in perturbations and varying it
with respect to $\zeta$, we obtain the following linear perturbation equation

$(a^{2}\epsilon\zeta^{\prime})^{\prime}-a^{2}\epsilon\,\partial^{2}\zeta=0\,,$
(3)

where a prime represents the derivative with respect to the conformal time
$\tau=\int a^{-1}dt$, and

$\epsilon=-\frac{\dot{H}}{H^{2}}\,,\qquad H=\frac{\dot{a}}{a}\,,$ (4)

with a dot being the derivative with respect to $t$. If we consider the scalar
field $\phi$ as an inflaton, the curvature perturbation exits the Hubble
horizon during inflation. Far outside the Hubble radius, the spatial gradient
term $\partial^{2}\zeta$ in Eq. (3) can be neglected relative to the first
term. In this regime, we obtain the following solution for the linear
curvature perturbation

$\zeta=c_{1}+c_{2}\int\frac{d\tau}{a^{2}\epsilon}\,,$ (5)

where $c_{1}$ and $c_{2}$ are integration constants. In standard slow-roll
inflation where $\epsilon$ slowly varies in time, the second term on the
right-hand side (RHS) of Eq. (5) corresponds to a decaying mode. In this case,
$\zeta$ approaches a constant $c_{1}$ after the Hubble radius crossing. We
note that, in the context of ultra-slow-roll inflation where there is a
temporal epoch in which $\epsilon$ rapidly decreases toward 0, the second term
in Eq. (5) can lead to the enhancement of $\zeta$.

In this paper, we are primarily interested in the nonlinear curvature
perturbation and its three-point correlation function in the context of slow-
roll inflation. For this purpose, the cubic-order action of $\zeta$, whose
explicit form will be presented in Sec. III, plays a crucial role. Such a
cubic action contains boundary terms as well as terms proportional to the
left-hand side of the linear EOM (3). Note that the latter is generally
nonvanishing for nonlinear perturbations. From the viewpoint of QFT, we need
to properly deal with these two contributions to the action for calculating
the three-point correlation function as well as loop corrections. In this
section, we will explain two general frameworks for computing the correlation
functions in the presence of the boundary and EOM terms.

###  II.1 The operator formalism

We begin with the operator formalism for dealing with the boundary terms in
the action. Note that the role of such contributions has been discussed in the
literature [8, 10, 11]. For concreteness, we consider the following
interaction Hamiltonian of curvature perturbations

$\displaystyle\hat{H}_{\rm int}=\hat{H}_{\rm bulk}+\int
d^{3}\bm{x}\left[({\cal{D}}_{2}\hat{\zeta})\hat{f}(\tau)+\frac{d}{d\tau}\hat{g}(\tau)\right]\,,$
(6)

where $\hat{H}_{\rm bulk}$ is the Hamiltonian in the bulk, $\hat{f}(\tau)$ and
$\hat{g}(\tau)$ are arbitrary time-dependent functions of the operator
$\hat{\zeta}$ possibly including derivative terms, and we have defined

${\cal{D}}_{2}\hat{\zeta}\equiv 2M_{\rm
Pl}^{2}\left[(a^{2}\epsilon\hat{\zeta}^{\prime})^{\prime}-a^{2}\epsilon\,\partial^{2}\hat{\zeta}\right]\,.$
(7)

The first contribution to the integrand in Eq. (6) is the EOM term and the
second one corresponds to the boundary term. Note that, in the spacetime that
has spatial translation invariance such as the FLRW spacetime, the spatial
total derivative terms are simply neglected without any subtlety thanks to the
three-momentum conservation.

In the operator formalism, the interaction picture operator
$\hat{\zeta}^{I}(\tau,\bm{x})$ consists of the creation and annihilation
operators multiplied by either positive or negative frequency solutions to the
free equation of motion ${\cal{D}}_{2}\zeta=0$. Therefore, we can drop the
contribution of EOM terms from the interaction Hamiltonian operator
independently of the function $\hat{f}(\tau)$. Thus, we will focus only on the
total time derivative term in the Hamiltonian. As we will discuss later, it is
worth emphasizing that this is not the case in path-integral formalism where
the derivatives appearing interactions act on the propagators.

Secondly, we should note that the total time derivative cannot be simply
integrated within the time-dependent perturbation theory. The time-evolution
operator is given by

$\displaystyle\hat{U}(\tau,\tau_{0})=T\exp\left[-{\rm
i}\int_{\tau_{0}}^{\tau}d\tilde{\tau}\hat{H}_{\rm
int}(\tilde{\tau})\right]\,,$ (8)

and one may think that the total time-derivative operator can be simply
integrated as

$\displaystyle\hat{U}(\tau,\tau_{0})\overset{?}{=}e^{-{\rm i}\int
d^{3}\bm{x}\hat{g}(\tau)}T\exp\left[-{\rm i}\int
d^{3}\bm{x}\int_{\tau_{0}}^{\tau}d\tilde{\tau}\hat{H}_{\rm
bulk}(\tilde{\tau})\right]e^{{\rm i}\int d^{3}\bm{x}\hat{g}(\tau_{0})}\,.$ (9)

However, the RHS is simply wrong unless $\hat{g}(\tau)$ commutes with bulk
terms $\hat{H}_{\rm bulk}(\tau)$. Thus, the total time-derivative term cannot
really be treated as the boundary term at the final surface444If the time
integration is naively carried out, the total time derivative terms can be
regarded as the operators at the “initial” and the “final” Cauchy slices.
Assuming that interactions are turned off at the initial time, we would be
left with the operators on the final surface. It is often assumed by the
Bunch-Davies vacuum condition (or ${\rm i}\epsilon$ prescription or Euclidean
vacuum condition).. The correct expression when $\hat{H}_{\rm bulk}(\tau)$ is
included is

$\displaystyle\hat{U}(\tau,\tau_{0})=$ $\displaystyle\hat{\bm{1}}-{\rm
i}\int_{\tau_{0}}^{\tau}d\tau_{1}\left[\hat{H}_{\rm
bulk}(\tau_{1})+\frac{d}{d\tau_{1}}\hat{g}(\tau_{1})\right]$
$\displaystyle+(-{\rm
i})^{2}\int^{\tau}_{\tau_{0}}d\tau_{1}\int^{\tau_{1}}_{\tau_{0}}d\tau_{2}\left[\hat{H}_{\rm
bulk}(\tau_{1})+\frac{d}{d\tau_{1}}\hat{g}(\tau_{1})\right]\left[\hat{H}_{\rm
bulk}(\tau_{2})+\frac{d}{d\tau_{2}}\hat{g}(\tau_{2})\right]+\cdots\,,$ (10)

where we have formally introduced the bulk interaction Hamiltonian and the
ellipses denote higher-order terms. Now, it is clear that the total time-
derivative terms cannot be simply integrated starting from the second order in
the interaction Hamiltonian. Nevertheless, at first order, one can perform the
time integration and obtain

$\displaystyle\hat{U}(\tau,\tau_{0})|_{\rm 1st}=\hat{\bm{1}}-{\rm
i}\hat{g}(\tau)-{\rm i}\int_{\tau_{0}}^{\tau}d\tau_{1}\hat{H}_{\rm
bulk}(\tau_{1}),$ (11)

where we have assumed $\hat{g}(\tau_{0})=0$, and the total time derivative
term contributes to the correlation function of operators
$\langle\hat{O}_{1}\hat{O}_{2}\rangle$ as

$\displaystyle\langle\hat{O}_{1}(\tau_{*})\hat{O}_{2}(\tau_{*})\rangle\sim\langle\hat{O}^{I}_{1}(\tau_{*})\hat{O}^{I}_{2}(\tau_{*})\rangle+{\rm
i}\int_{\tau_{0}}^{\tau_{*}}d\tau\langle[\hat{H}^{I}_{\rm
bulk}(\tau),\hat{\mathcal{O}}^{I}_{1}(\tau_{*})\hat{\mathcal{O}}^{I}_{2}(\tau_{*})]\rangle+{\rm
i}\langle[\hat{g}(\zeta^{I}(\tau_{*})),\hat{O}^{I}_{1}(\tau_{*})\hat{O}^{I}_{2}(\tau_{*})]\rangle,$
(12)

at first order in the interaction Hamiltonian (the operator with “$I$” denotes
the interaction picture operator). We emphasize that such a simple expression
is not available at least from the second order in the interaction Hamiltonian
as mentioned above.

Let us illustrate how the boundary Hamiltonian plays its role in the
computation of correlation functions. We consider a simple boundary
Hamiltonian $\hat{g}(\tau)=\int
d^{3}\bm{x}A(\tau)\hat{\zeta}(\tau,\bm{x})\hat{\pi}_{\zeta}(\tau,\bm{x})\hat{\zeta}(\tau,\bm{x})$,
where $A(\tau)$ is a real function and $\hat{\pi}_{\zeta}$ denotes the
canonical conjugate momentum of $\hat{\zeta}$. For a three-point correlation
function
$\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})\rangle$
we find, from the last term of Eq. (12),

$\displaystyle\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})\rangle$
$\displaystyle\supset$ $\displaystyle{\rm i}A(\tau_{*})\int
d^{3}\bm{x}\langle[\hat{\zeta}(\tau_{*},\bm{x})\hat{\pi}_{\zeta}(\tau_{*},\bm{x})\hat{\zeta}(\tau_{*},\bm{x}),\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})]\rangle$
(13) $\displaystyle=$ $\displaystyle{\rm i}A(\tau_{*})\int
d^{3}\bm{x}\langle\hat{\zeta}(\tau_{*},\bm{x})[\hat{\pi}_{\zeta}(\tau_{*},\bm{x}),\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})]\hat{\zeta}(\tau_{*},\bm{x})\rangle$
$\displaystyle=$ $\displaystyle
A(\tau_{*})\langle\hat{\zeta}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})\hat{\zeta}^{2}(\tau_{*},\bm{x}_{1})+({\rm
cyclic})\rangle$ $\displaystyle\to$ $\displaystyle
A(\tau_{*})\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\rangle\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{3})\rangle+(\rm
cyclic)\,,$

where “$(\rm cyclic)$” denotes the cyclic permutation with respect to
$(1,2,3)$ and the last line shows only the connected part of the correlation
function. Here, we have used the canonical commutation relation in the third
line, from which we find the following: The equal time commutation relation
yields the spatial $\delta$-function, and an operator “contracted” by the
commutation relation is effectively “redefined”. This behavior is nothing but
the field redefinition prescription proposed by Maldacena [7]. Indeed, we
could reach the same expression by performing the field redefinition
$\zeta\to\zeta+A\zeta^{2}$. This point has been pointed out in Refs. [8, 10,
11]. As we will see in the next section, the role of the “boundary terms” is
played instead by EOM terms within the path-integral formalism.

###  II.2 The path-integral formalism

In this subsection, we clarify how the boundary and the EOM terms contribute
to the evaluation of the cosmological correlation functions in the path-
integral formalism, which looks different from that in the operator formalism.
In the operator approach, we take the expectation value of interaction picture
operators that possibly contain derivatives, whereas, in the path-integral
approach, correlation functions consist of free propagators, and the
derivatives contained in the interaction vertices would act from outside of
the propagator. In other words, we do not evaluate operators with derivatives,
but differentiate propagators when vertices contain differential operators.
Then, we cannot ignore the terms proportional to the EOM as the (anti) time-
ordered products do not vanish on-shell, while non-time-ordered products do.
Therefore, in the Schwinger-Keldysh path integral, the EOM term should not be
neglected.

Let us briefly review the Schwinger-Keldysh path-integral formalism. See e.g.,
Refs. [6, 31] for detailed discussions. As we evaluate expectation values
rather than S-matrix elements, there are “bra” and “ket” fields555They are
necessary for anti-time ordered and time ordered evolution associated with bra
and ket vectors. $\zeta^{-}$ and $\zeta^{+}$, respectively, and the degrees of
freedom are doubled in comparison to the standard Feynman path integral. The
free generating functional of the Schwinger-Keldysh path integral is given by

$\displaystyle
Z_{0}[J^{+},J^{-}]=\mathcal{N}_{0}\int{\mathcal{D}}\zeta^{+}\mathcal{D}\zeta^{-}\delta(\zeta^{+}(\tau_{f},\bm{x})-\zeta^{-}(\tau_{f},\bm{x}))\exp\left[-{\rm
i}S_{2}[\zeta^{-}]+{\rm i}S_{2}[\zeta^{+}]+{\rm i}\int d\tau
d^{3}{\bm{x}}(J^{+}\zeta^{+}-J^{-}\zeta^{-})\right],$ (14)

where $\tau_{f}$ denotes the final slice time, $J^{\pm}$ are source functions,
and $\mathcal{N}_{0}$ is a normalization factor, and $S_{2}[\zeta]$ denotes
the quadratic action of $\zeta$ given by

$S_{2}[\zeta]=M_{\rm Pl}^{2}\int d\tau
d^{3}{\bm{x}}\,a^{2}\epsilon\left[\zeta^{\prime
2}-(\partial\zeta)^{2}\right]\,.$ (15)

The $\delta$-function enforces the field configurations of $\zeta^{\pm}$ to
coincide at $\tau=\tau_{f}$. The partition function of the interacting theory
can be perturbatively calculated by

$\displaystyle Z[J^{+},J^{-}]=\mathcal{N}\exp\left[{\rm i}S_{\rm
int}\left[\frac{\delta}{{\rm i}\delta J^{+}}\right]-{\rm i}S_{\rm
int}\left[\frac{\delta}{-{\rm i}\delta
J^{-}}\right]\right]Z_{0}[J_{+},J_{-}]\,,$ (16)

where $S_{\rm int}$ denotes the interacting part of the action possibly
including the spacetime derivative of $\zeta$. The connected correlation
function can be computed via the generating functional $W[J^{+},J^{-}]=-{\rm
i}\log\left(Z[J^{+},J^{-}]\right)$, with which the three-point correlation
function, for example, is given by

$\displaystyle\langle\zeta^{+}(\tau_{*},\bm{x}_{1})\zeta^{+}(\tau_{*},\bm{x}_{2})\zeta^{+}(\tau_{*},\bm{x}_{3})\rangle=\frac{\delta^{3}}{{\rm
i}^{3}\delta J^{+}(\tau_{*},\bm{x}_{1})\delta J^{+}(\tau_{*},\bm{x}_{2})\delta
J^{+}(\tau_{*},\bm{x}_{3})}W[J^{+},J^{-}]\biggr{|}_{J=0}\,.$ (17)

The evaluation of the correlation functions can be done by contracting the
interaction vertices and the operators put at $\tau=\tau_{*}$, and due to the
presence of the $\pm$-fields, we have four different propagators:

$\displaystyle G^{++}(\tau_{1},\bm{x}_{1};\tau_{2},\bm{x}_{2})=$
$\displaystyle\frac{\delta^{2}}{{\rm i}^{2}J^{+}(\tau_{1},\bm{x}_{1})\delta
J^{+}(\tau_{2},\bm{x}_{2})}Z_{0}[J^{+},J^{-}]\biggr{|}_{J=0}=\langle
T(\hat{\zeta}(\tau_{1},\bm{x}_{1})\hat{\zeta}(\tau_{2},\bm{x}_{2}))\rangle.$
(18) $\displaystyle G^{+-}(\tau_{1},\bm{x}_{1};\tau_{2},\bm{x}_{2})=$
$\displaystyle\frac{\delta^{2}}{J^{+}(\tau_{1},\bm{x}_{1})\delta
J^{-}(\tau_{2},\bm{x}_{2})}Z_{0}[J^{+},J^{-}]\biggr{|}_{J=0}=\langle\hat{\zeta}(\tau_{2},\bm{x}_{2})\hat{\zeta}(\tau_{1},\bm{x}_{1})\rangle,$
(19) $\displaystyle G^{-+}(\tau_{1},\bm{x}_{1};\tau_{2},\bm{x}_{2})=$
$\displaystyle\frac{\delta^{2}}{J^{-}(\tau_{1},\bm{x}_{1})\delta
J^{+}(\tau_{2},\bm{x}_{2})}Z_{0}[J^{+},J^{-}]\biggr{|}_{J=0}=\langle\hat{\zeta}(\tau_{1},\bm{x}_{1})\hat{\zeta}(\tau_{2},\bm{x}_{2})\rangle,$
(20) $\displaystyle G^{--}(\tau_{1},\bm{x}_{1};\tau_{2},\bm{x}_{2})=$
$\displaystyle\frac{\delta^{2}}{(-{\rm i})^{2}J^{-}(\tau_{1},\bm{x}_{1})\delta
J^{-}(\tau_{2},\bm{x}_{2})}Z_{0}[J^{+},J^{-}]\biggr{|}_{J=0}=\langle\bar{T}(\hat{\zeta}(\tau_{1},\bm{x}_{1})\hat{\zeta}(\tau_{2},\bm{x}_{2}))\rangle\,,$
(21)

where $T(\bar{T})$ is (anti-)time ordering symbol.

One question arises here. How should the time $\tau_{f}$ be chosen for the
final slice, which is introduced to calculate the expectation value in the
path-integral formalism? As long as $\tau_{f}\geq\tau_{*}$ is satisfied, the
choice is arbitrary and should not affect the final result. In general,
$\tau_{f}=\tau_{*}$ has been the preferred choice. This is because, since we
are mostly interested in correlation functions of $\zeta$ without derivatives
at $\tau_{*}$, we can freely take them to be either $+$ or $-$ fields thanks
to the $\delta$-function in the partition function. However, the
$\delta$-function $\delta(\zeta^{+}(\tau_{f})-\zeta^{-}(\tau_{f}))$ that
enforces the bra- and ket-fields $\zeta^{\mp}$ to be equal is not necessary by
the following reason: In cosmological applications, we are mostly interested
in the equal time correlation functions of the same operator
$\langle\zeta(\tau_{*},\bm{x}_{1})\cdots\zeta(\tau_{*},\bm{x}_{n})\rangle$. In
such cases, they are products of commuting operators and their ordering does
not really change anything. Thus, they can be identified as either $+$ or $-$
fields666Even if one wants to evaluate the expectation value of a product of
operators that are not commutative, the ordering ambiguity does not contribute
to the connected diagram.. Therefore, in the following, we choose
$\tau_{f}>\tau_{*}$, which makes unambiguous the contributions from the EOM
terms and the boundary terms. As we will see later, the EOM terms lead to a
$\delta$-function, $\delta(\tau-\tau_{*})$, so it is more healthy to take the
integral range of $\tau$ as fully encompassing $\tau_{*}$.

Now, let us consider the following action

$\displaystyle S_{\rm int}=-\int d\tau
d^{3}\bm{x}\left[(\mathcal{D}_{2}\zeta)f(\zeta)+\frac{d}{d\tau}g(\zeta)\right]\,,$
(22)

which describes the interaction. One of the most important points is that
$\mathcal{D}_{2}$ annihilates the mixed propagators $G^{+-}$ and $G^{-+}$, but
it does not annihilate (anti-)time ordered one $G^{++(--)}$. Thus, we find
that $\mathcal{D}_{2}G^{++}(\tau,\bm{x};\tau_{1},\bm{x}_{1})=-{\rm
i}\delta(\tau-\tau_{1})\delta^{3}(\bm{x}-\bm{x}_{1})$, with the understanding
that $\mathcal{D}_{2}$ acts on either $(\tau,\bm{x})$ or
$(\tau_{1},\bm{x}_{1})$. The Hermite conjugate of it yields that of the anti-
time ordered products. Suppose we consider three-point correlation function
$\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})\rangle$
and treat them as $\zeta^{+}$. For concreteness, we formally write777One may
consider the case where there are derivatives on $\zeta$, and even in such
cases the following observations are applied. $f(\zeta)=\lambda\zeta^{2}$. At
first order in the interaction vertices, we would have contributions to the
three-point function from the EOM term as

$\displaystyle\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})\rangle\supset$
$\displaystyle-2{\rm i}\lambda\int d\tau
d^{3}\bm{x}\,\mathcal{D}_{2}G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{1})G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{2})G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{3})$
$\displaystyle+2{\rm i}\lambda\int d\tau
d^{3}\bm{x}\,\mathcal{D}_{2}G^{-+}(\tau,\bm{x};\tau_{*},\bm{x}_{1})G^{-+}(\tau,\bm{x};\tau_{*},\bm{x}_{2})G^{-+}(\tau,\bm{x};\tau_{*},\bm{x}_{3})+(\text{cyclic})$
$\displaystyle=$ $\displaystyle-2\lambda\int d\tau
d^{3}\bm{x}\,\delta(\tau-\tau_{*})\delta(\bm{x}-\bm{x}_{1})G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{2})G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{3})+(\text{cyclic})$
$\displaystyle=$
$\displaystyle-2\lambda\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\rangle\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{3})\rangle+(\text{cyclic}),$
(23)

where “(cyclic)” means the terms with cyclic permutation of the indices
$1,2,3$, and we have used the properties of the propagators mentioned above.
Thus the EOM terms do not vanish, unlike the operator formalism. We also find
the correspondence to the field redefinition method proposed by Maldacena [7].
He proposed to introduce a new variable $\zeta=\zeta_{n}-f(\zeta_{n})$, by
which the quadratic part of the action cancels the EOM term888We note that the
sign difference of $\zeta=\zeta_{n}-f(\zeta_{n})$ in comparison to Ref. [7]
arises from the fact that we have defined $\delta
S_{2}/\delta\zeta=\mathcal{D}_{2}\zeta$, which seems opposite to Ref. [7]. .
However, from the observation here, we find that the EOM terms themselves play
a role of the field redefinition at least at first order in the interaction
vertices. It might be worth noting that the field redefinition in general
leads to nontrivial Jacobian factor in the path integral, and therefore, such
a technique might be cumbersome, whereas leaving the EOM terms may be tedious
but straightforward999Nevertheless, we expect that any methods would be
equally complicated starting from the second order in interaction vertices..

We note the EOM terms always yield the time $\delta$-function that “brings the
interaction vertices to the same time plane on which operators are inserted”.
Therefore, we have to be careful about the time-ordered product and the
Heaviside step function $\theta(\tau)$ in the propagators (21). We will take
the following convention

$\displaystyle\theta(\tau)=\left\\{\begin{array}[]{ll}1&\text{for }\tau>0\\\
1/2&\text{for }\tau=0\\\ 0&\text{for }\tau<0\end{array}\right.\,.$ (27)

This prescription keeps the reality of the correlation function of real
fields101010Independently of whether inserted operators are treated as $+$ or
$-$ fields, the $\pm{\rm i}$ from the vertices precisely cancel the $\mp{\rm
i}$ associated with $\mathcal{D}_{2}G^{++(--)}(x-x^{\prime})=-(+){\rm
i}\delta^{4}(x-x^{\prime})$. Therefore, either choice yields the same result
and the resultant contribution becomes real with the above prescription to the
Heaviside $\theta$-function..

Let us turn to the discussion on the boundary terms. As we mentioned before,
we set the final slice time $\tau_{f}$ to be later than $\tau_{*}$. For
concreteness, we assume $g(\zeta)=\lambda_{2}\zeta^{\prime}\zeta^{2}$. Then,
we find

$\displaystyle\langle\hat{\zeta}(\tau_{*},\bm{x}_{1})\hat{\zeta}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})\rangle$
$\displaystyle\supset-2{\rm i}\int
d^{3}\bm{x}\biggl{[}\partial_{\tau}G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{1})G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{2})G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{3})$
$\displaystyle\qquad-\partial_{\tau}G^{-+}(\tau,\bm{x};\tau_{*},\bm{x}_{1})G^{-+}(\tau,\bm{x};\tau_{*},\bm{x}_{2})G^{-+}(\tau,\bm{x};\tau_{*},\bm{x}_{3})+(\text{cyclic})\biggr{]}_{\tau\to\tau_{f}}$
$\displaystyle=-2{\rm i}\int
dK\delta^{3}\left(\sum_{i=1}^{3}\bm{k}_{i}\right)e^{{\rm
i}(\sum_{i=1}^{3}\bm{k}_{i}\cdot\bm{x}_{i})}\biggl{[}\partial_{\tau}G^{++}_{k_{1}}(\tau,\tau_{*})G^{++}_{k_{2}}(\tau,\tau_{*})G^{++}_{k_{3}}(\tau,\tau_{*})$
$\displaystyle\quad-\partial_{\tau}G^{-+}_{k_{1}}(\tau,\tau_{*})G^{-+}_{k_{2}}(\tau,\tau_{*})G^{-+}_{k_{3}}(\tau,\tau_{*})+(\text{cyclic})\biggr{]}_{\tau\to\tau_{f}},$
(28)

where $\int dK=(2\pi)^{-6}\int d^{3}\bm{k}_{1}d^{3}\bm{k}_{2}d^{3}\bm{k}_{3}$.
Since
$G^{++}_{k_{i}}(\tau_{f},\tau_{*})=G^{-+}_{k_{i}}(\tau_{f},\tau_{*})=\zeta_{k_{i}}(\tau_{f})\bar{\zeta}_{k_{i}}(\tau_{*})$
for $\tau_{f}>\tau_{*}$, we find that the RHS of the above vanishes by
cancellation between $++$ and $-+$, namely

$\displaystyle\text{RHS of \eqref{PIB}}=0\,.$ (29)

Here, we should emphasize that we have taken the inserted $\zeta$ operators as
$+$-fields. Nevertheless, one can see that the cancellation occurs similarly
even if we treat them to be $-$-fields111111Essentially, in our prescription,
the boundary interactions appear at either the most left or the most right for
$\pm$-fields, and therefore, the $\pm$-vertices are not distinguished except
for the overall sign difference, by which the contributions from
$\pm$-Lagrangian cancel to each other.. In this prescription, the boundary
terms play no roles unlike the operator formalism where the EOM terms simply
vanish. This result is expected from the causality since the interaction on
the future boundary must not affect the expectation values of operators on
$\tau_{*}<\tau_{f}$. As we will see in Sec. III, the above prescription
reproduces consistency relations of correlation functions of curvature
perturbations.

Note however that there remains a puzzle: If we naively put the operators at
the boundary time, i.e., $\tau_{*}=\tau_{f}$, the above cancellation seems not
to occur and there would remain some nonvanishing contribution. Such a
subtlety may arise because the boundary interaction appears at precisely the
same time as the operators we evaluate. We will not discuss such a technical
issue further as our prescription does not give such an ambiguity.

Let us comment on the boundary time in the operator formalism. In this
formalism, the time evolution operator appears by translating the Heisenberg
operator by the interaction picture operators. More specifically, if one wants
to evaluate $\langle\hat{O}(\tau_{*})\rangle$, where $\hat{O}(\tau_{*})$ is
supposed to be the Heisenberg picture operator, we rewrite it as

$\displaystyle\hat{O}(\tau_{*})=\hat{U}^{\dagger}(\tau_{*},\tau_{0})\hat{O}^{I}(\tau_{*})\hat{U}(\tau_{*},\tau_{0}).$
(30)

This implies that the “boundary time” is fixed by the operators that we
evaluate. This feature differs from the path-integral case as we put operators
by differentiating generating functionals with respect to the source
functions, and therefore the “boundary time” is arbitrary. Such a difference
may explain why the boundary terms are not taken to future infinity in the
operator formalism121212Of course, if we evaluate the operators at future
infinity, this is not the case, and boundary operators also appear at infinity
at least at the tree level..

##  III Example: three-point correlation functions and consistency relations
in slow-roll inflation

In this section, we would like to check what the boundary and the EOM terms
play the role in correlation functions. In particular, we focus on the
consistency relations of curvature perturbations, which are important in
proving the absence of the time-dependence of $\zeta$ that exits the Hubble
horizon beyond a tree-level approximation [32].

We assume a single-clock inflation model given by the action (1) and take into
account the Gibbons-Hawking-York action $S_{\rm GHY}$ [33, 34]. The purpose of
incorporating the term $S_{\rm GHY}$ is to make the variational principle
well-defined in the bounded four-dimensional spacetime and to obtain the
Lagrangian to be functions of $\zeta$ and $\zeta^{\prime}$ (not including
$\zeta^{\prime\prime}$). In the unitary gauge, the cubic-order action of
curvature perturbations consists of the following set of terms131313In the
following discussion, we focus only on the cubic terms. Strictly speaking,
there are quadratic total time derivative terms, which seem to contribute as
boundary terms. Nevertheless, such terms generally cancel when one converts
the Lagrangian to the Hamiltonian. Therefore, one may safely neglect such
terms. (see e.g., Ref. [35] for detailed derivation):

$\displaystyle S_{\rm bulk}=$ $\displaystyle M_{\rm Pl}^{2}\int d\tau
d^{3}\bm{x}\,a^{2}\biggl{[}\epsilon^{2}\left(1-\frac{1}{2}\epsilon\right)\zeta(\zeta^{\prime})^{2}+\epsilon^{2}\zeta(\partial\zeta)^{2}+\frac{1}{2}\eta^{\prime}\epsilon\zeta^{\prime}\zeta^{2}-2\epsilon\zeta^{\prime}\partial_{k}\zeta\partial_{k}(\partial^{-2}\zeta^{\prime})+\frac{1}{2}\epsilon^{3}\zeta(\partial_{k}\partial_{l}\partial^{-2}\zeta^{\prime})^{2}\biggr{]},$
(31) $\displaystyle S_{\partial}=$ $\displaystyle M_{\rm Pl}^{2}\int
d^{3}\bm{x}\,\biggl{[}-a^{2}\epsilon\eta\zeta^{\prime}\zeta^{2}-\frac{a\epsilon}{H}\zeta(\zeta^{\prime})^{2}-\epsilon^{2}\zeta^{\prime}\zeta\partial^{2}\zeta+2a\epsilon^{3}\zeta^{2}\partial^{2}\zeta$
$\displaystyle\qquad+\frac{\epsilon}{2H^{2}}\zeta\partial_{k}\partial_{l}\zeta(\partial_{k}\partial_{l}\partial^{-2}\zeta^{\prime})-\frac{1}{6aH}\zeta\\{(\partial_{k}\partial_{l}\zeta)^{2}-(\partial^{2}\zeta)^{2}\\}-\frac{a\epsilon^{2}}{2H}\zeta(\partial_{k}\partial_{l}\partial^{-2}\zeta^{\prime})^{2}\biggr{]}\Biggr{|}_{\tau=\tau_{f}},$
(32) $\displaystyle S_{\rm EOM}=$ $\displaystyle\int d\tau
d^{3}\bm{x}\,\mathcal{D}_{2}\zeta\biggl{[}\frac{1}{4}\eta\zeta^{2}+\frac{1}{aH}\zeta^{\prime}\zeta-\frac{1}{4(aH)^{2}}\\{(\partial\zeta)^{2}-\partial^{-2}\partial_{k}\partial_{l}(\partial_{k}\zeta\partial_{l}\zeta)\\}$
$\displaystyle+\frac{\epsilon}{aH}\\{\partial_{k}\zeta\partial_{k}\partial^{-2}\zeta^{\prime}-\partial^{-2}\partial_{k}\partial_{l}(\partial_{k}\zeta\partial_{l}\partial^{-2}\zeta)\\}\biggr{]}\,,$
(33)

where $S_{\rm bulk}$, $S_{\partial}$, and $S_{\rm EOM}$ correspond to the
contributions of the bulk, boundary, and EOM terms, respectively, and

$\eta=\frac{\dot{\epsilon}}{H\epsilon}\,.$ (34)

Note that the nonlocal operator $\partial^{-2}$ is understood in the sense of
Fourier transform.

###  III.1 Path-integral viewpoint

####  III.1.1 $\langle\zeta\zeta\zeta\rangle$

Let us compute a three-point function
$\langle\zeta(\tau_{*},\bm{x}_{1})\zeta(\tau_{*},\bm{x}_{2})\zeta(\tau_{*},\bm{x}_{3})\rangle$
in the path-integral formalism. As described in Sec. II.2, the boundary terms
$S_{\partial}$ do not contribute to the tree-level correlation functions.
Therefore, we are left with the bulk and the EOM terms. From the general
consideration in Sec. II.2, at first order, the contraction with
$\mathcal{D}_{2}\zeta$ and one of $\zeta(\tau_{*},\bm{x}_{i})$ ($i=1,2,3$)
yields (anti-)time-ordered propagator that gives a $\delta$-function
$\delta(\tau-\tau_{*})\delta^{3}(\bm{x}-\bm{x}_{i})$ by $\mathcal{D}_{2}$
acting on it, and the rest of $\zeta_{j\neq i}$ are contracted with remaining
fields in the EOM terms, which reproduces the known result. More specifically,
by taking $\tau_{*}\to 0$ where $a\to\infty$, only the EOM term $S_{\rm
EOM}\to\int d\tau
d^{3}\bm{x}\,\mathcal{D}_{2}\zeta\left(\eta\zeta^{2}/4\right)$
survives141414It is easy to confirm it even after taking contraction
explicitly.. Then, we find the contribution from the EOM term to be

$\displaystyle\langle\zeta(\tau_{*},\bm{x}_{1})\zeta(\tau_{*},\bm{x}_{2})\zeta(\tau_{*},\bm{x}_{3})\rangle\biggr{|}_{\text{EOM}}$
$\displaystyle\underset{\tau_{*}\to 0}{\sim}$ $\displaystyle+{\rm i}\int d\tau
d^{3}\bm{x}\frac{\eta}{2}\mathcal{D}_{2}G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{1})G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{2})(\tau,\bm{x};\tau_{*},\bm{x}_{3})+({\rm
cyclic})$ (35) $\displaystyle=$
$\displaystyle\frac{\eta(\tau_{*})}{2}G^{++}(\tau_{*},\bm{x}_{1};\tau_{*},\bm{x}_{2})G^{++}(\tau_{*},\bm{x}_{1};\tau_{*},\bm{x}_{3})+({\rm
cyclic})\,,$

where we have used
$\mathcal{D}_{2}G^{++}(\tau,\bm{x};\tau_{*},\bm{x}_{1})=-{\rm
i}\delta(\tau-\tau_{*})\delta^{3}(\bm{x}-\bm{x}_{1})$ and formally written the
time-dependence of $\eta$ explicitly. We here emphasize again that there is no
need for field redefinition, but we just straightforwardly compute the
correlation function. The $-$ action yields just a vanishing contribution.
Note that, if we assume $\zeta^{-}(\tau_{*},\bm{x}_{i})$ to be inserted at the
boundary, the $-$ action instead contributes and the anti-time ordered
propagator $G^{--}$ yields the same $\delta$ function with an opposite
imaginary unit $+{\rm i}\delta(\tau-\tau_{*})\delta^{3}(\bm{x}-\bm{x}_{i})$.
This yields exactly the same result. The bulk contribution is computed in the
same way as the known result studied in the literature.

To check the behavior of the consistency relation for the wavenumbers
$k_{3}\ll k_{1}\sim k_{2}$, we show the final result expanded in terms of the
slow roll parameters $\epsilon$ and $\eta$ (leaving $\tau_{*}<0$) in Fourier
space, where we take the convention $\zeta(\tau,\bm{k})\equiv\int
d^{3}\bm{x}\,e^{{\rm i}\bm{k}\cdot\bm{x}}\zeta(\tau,\bm{x})$. Having fully
computed the first-order contributions from the bulk and the EOM
interactions151515The boundary terms do not contribute as explained below (28)
and hence need not be taken account of. shown above to the three-point
function and performed the Fourier transformation, we reach the result in the
squeezed limit $k_{3}\ll k_{1}\sim k_{2}$ as

$\displaystyle\langle\zeta(\tau_{*},\bm{k}_{1})\zeta(\tau_{*},\bm{k}_{2})\zeta(\tau_{*},\bm{k}_{3})\rangle_{c}\biggr{|}_{\text{tot}}\underset{k_{3}\ll
k_{1}\sim
k_{2}}{\longrightarrow}-\frac{H^{4}k_{1}^{2}\tau_{*}^{2}(2-\eta)}{16k_{3}^{3}k_{1}^{3}M_{\rm
Pl}^{6}\epsilon^{2}}+\frac{H^{4}(\eta+2\epsilon)}{16k_{3}^{3}k_{1}^{3}M_{\rm
Pl}^{6}\epsilon^{2}}+\mathcal{O}(k_{3}^{-2})\,,$ (36)

where $\langle\ \cdot\ \rangle_{c}$ denotes the correlation function without
overall $\delta$-function $(2\pi)^{3}\delta(\sum_{i}\bm{k}_{i})$ and we have
only shown the leading and next-to-leading order terms in slow-roll
parameters. We note that the power spectrum under the slow-roll approximation
is given by

$\displaystyle\langle{\zeta}(\tau_{*},\bm{k}){\zeta}(\tau_{*},-\bm{k})\rangle_{c}\approx\left\\{\begin{array}[]{ll}\dfrac{H^{2}}{8k^{3}M_{\rm
Pl}^{2}\epsilon}(-k\tau_{*})^{-\eta-2\epsilon}&~{}~{}\text{for }k\tau_{*}\to
0\,,\\\ \dfrac{H^{2}k^{2}\tau_{*}^{2}}{4k^{3}M_{\rm
Pl}^{2}\epsilon}&~{}~{}\text{for }k\tau_{*}\gg 1\,,\end{array}\right.$ (39)

where we have used the exact mode function

$\zeta(\tau,k)=-\frac{\sqrt{\pi}}{2\sqrt{2}\epsilon M_{\rm
Pl}a}\sqrt{-\tau}H_{\nu}(-k\tau)\,,$ (40)

with $\nu=3/2+\epsilon+\eta/2$. Then, we find

$\displaystyle\langle\zeta(\tau_{*},\bm{k}_{1})\zeta(\tau_{*},\bm{k}_{2})\zeta(\tau_{*},\bm{k}_{3})\rangle_{c}\biggr{|}_{\text{tot}}\approx-\frac{d\log(k_{1}^{3}\langle{\zeta}(\tau_{*},\bm{k}_{1}){\zeta}(\tau_{*},-\bm{k}_{1})\rangle_{c})}{d\log
k_{1}}\langle{\zeta}(\tau_{*},\bm{k}_{3}){\zeta}(\tau_{*},-\bm{k}_{3})\rangle_{c}\langle{\zeta}(\tau_{*},\bm{k}_{1}){\zeta}(\tau_{*},-\bm{k}_{1})\rangle_{c}\,,$
(41)

which holds for $k_{3}\ll k_{1}\sim k_{2}$ and for either $k_{1}\tau_{*}\gg 1$
or $k_{1}\tau_{*}\ll 1$. This is the well-known Maldacena’s consistency
relation [7]. It would be noteworthy that the leading-order contribution to
the above for the sub-Hubble mode $k_{1}\tau_{*}\gg 1$ comes from the EOM term
rather than the bulk terms.

####  III.1.2 $\langle\dot{\zeta}\dot{\zeta}\zeta\rangle$

Let us consider another nontrivial example of the three-point correlation
function. It would be illustrative to consider the consistency relation of
$\langle\dot{\hat{\zeta}}(\tau_{*},\bm{k}_{1})\dot{\hat{\zeta}}(\tau_{*},\bm{k}_{2})\hat{\zeta}(\tau_{*},\bm{k}_{3})\rangle$.
Since we expect that the correlation function simply vanishes when
$\tau_{*}\to 0$ or all the modes are in the super-Hubble regime, we only focus
on the case $k_{3}\ll k_{1}\sim k_{2}$ while $k_{1}\tau_{*}=\text{finite}$.
The computation of the bulk contribution is rather straightforward since
nothing really changes from the standard calculation except that the inserted
operators contain extra time derivatives. Hence we focus only on the
contribution from EOM terms. If one notices that the tree-level contribution
of the EOM terms can be reproduced by the well-known field redefinition
procedure, the computation is also very straightforward. Nevertheless, with
the application to the higher-order corrections including loops in mind, we
give some details of the computation leaving the EOM terms as they are.
Firstly, we start with the position-space correlation function
$\langle\dot{\hat{\zeta}}(\tau_{*},\bm{x}_{1})\dot{\hat{\zeta}}(\tau_{*},\bm{x}_{2})\hat{\zeta}(\tau_{*},\bm{x}_{3})\rangle$
instead. The contractions yield three different contributions
$F_{1},F_{2},F_{3}$, which are given, respectively, by161616$F_{i}$’s
correspond to the contributions from the contraction of $\mathcal{D}_{2}\zeta$
in the EOM interaction and the external field $\zeta(\bm{x}_{i})$.

$\displaystyle F_{1}=$ $\displaystyle\frac{{\rm i}}{a^{2}(\tau_{*})}\int d\tau
d^{3}\bm{x}\,(-{\rm
i})\partial_{\tau_{2}}\delta(\tau-\tau_{2})\delta^{3}(\bm{x}-\bm{x}_{1})\biggl{[}\frac{\eta}{2}\partial_{\tau_{1}}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})G^{++}(\tau_{1},\bm{x}_{3};\tau,\bm{x})$
$\displaystyle+\frac{1}{aH}\partial_{\tau}\\{\partial_{\tau_{1}}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})G^{++}(\tau^{\prime},\bm{x}_{3};\tau,\bm{x})\\}-\frac{1}{2(aH)^{2}}\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\partial_{k}G^{++}(\tau_{1},\bm{x}_{3};\tau,\bm{x})$
$\displaystyle+\frac{1}{2(aH)^{2}}\partial^{-2}\partial_{k}\partial_{l}\\{\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\partial_{l}G^{++}(\tau_{1},\bm{x}_{3};\tau,\bm{x})\\}$
$\displaystyle-\frac{\epsilon}{aH}\partial^{-2}\partial_{k}\partial_{l}\partial_{\tau}\\{\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\partial_{l}G^{++}(\tau_{1},\bm{x}_{3};\tau,\bm{x})\\}$
$\displaystyle+\frac{\epsilon}{aH}\\{\partial_{\tau_{1}}\partial_{k}\partial^{-2}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\partial_{k}G^{++}(\tau_{1},\bm{x}_{3};\tau,\bm{x})+\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\partial_{k}\partial^{-2}G^{++}(\tau_{1},\bm{x}_{3};\tau,\bm{x})\\}\biggr{]}_{\tau_{1,2}\to\tau_{*}},$
(42) $\displaystyle F_{2}=F_{1}|_{\bm{x}_{1}\leftrightarrow\bm{x}_{2}},$ (43)

and

$\displaystyle F_{3}=$ $\displaystyle\frac{{\rm i}}{a^{2}(\tau_{*})}\int d\tau
d^{3}\bm{x}(-{\rm
i})\delta(\tau-\tau_{1})\delta^{3}(\bm{x}-\bm{x}_{3})\biggl{[}\frac{\eta}{2}\partial_{\tau_{1}}G^{++}(\tau_{1},\bm{x}_{1};\tau,\bm{x})\partial_{\tau_{1}}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})$
$\displaystyle+\frac{1}{aH}\partial_{\tau}\\{\partial_{\tau_{1}}G^{++}(\tau_{1},\bm{x}_{1};\tau,\bm{x})\partial_{\tau_{1}}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\\}-\frac{1}{2(aH)^{2}}\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{1};\tau,\bm{x})\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})$
$\displaystyle+\frac{1}{2(aH)^{2}}\partial^{-2}\partial_{k}\partial_{l}\\{\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{1};\tau,\bm{x})\partial_{\tau_{1}}\partial_{l}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\\}$
$\displaystyle+\frac{\epsilon}{aH}\\{\partial_{\tau_{1}}\partial_{k}\partial^{-2}G^{++}(\tau_{1},\bm{x}_{1};\tau,\bm{x})\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})+\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{1};\tau,\bm{x})\partial_{\tau_{1}}\partial_{k}\partial^{-2}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\\}$
$\displaystyle-\frac{\epsilon}{aH}\partial^{-2}\partial_{k}\partial_{l}\partial_{\tau}\\{\partial_{\tau_{1}}\partial_{k}G^{++}(\tau_{1},\bm{x}_{1};\tau,\bm{x})\partial_{\tau_{1}}\partial_{l}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})\\}\biggr{]}_{\tau_{1}\to\tau_{*}}.$
(44)

We have shown (possibly unnecessary) whole terms just for completeness. It is
straightforward to perform the integrals in each quantity, but only the $\tau$
integration in $F_{1,2}$ needs a little care due to the derivative acting on
the $\delta$-function. Since the $\tau_{2}$ derivative is acting on
$\delta$-function, we obtain

$\displaystyle
F_{1}=+\partial_{\tau}\biggl{[}\frac{\eta}{2}\partial_{\tau_{1}}G^{++}(\tau_{1},\bm{x}_{2};\tau,\bm{x})G^{++}(\tau_{1},\bm{x}_{3};\tau,\bm{x})+\cdots\biggr{]}\,.$
(45)

One may check that this is consistent with the result of field redefinition.
Thus, even if inserted operators contain derivatives, one can
straightforwardly compute correlation functions without the field redefinition
procedure.

After straightforward but messy calculations, it turns out that the bulk
contribution only gives the subleading contribution in the squeezing limit
$k_{3}\ll k_{1}\sim k_{2}$, and a specific EOM term in the Lagrangian,
${\cal{L}}_{\rm EOM}\supset({\cal{D}}_{2}\zeta)\zeta^{\prime}\zeta/aH$ yields
the leading contribution

$\displaystyle\langle\dot{\hat{\zeta}}(\tau_{*},\bm{k}_{1})\dot{\hat{\zeta}}(\tau_{*},\bm{k}_{2})\hat{\zeta}(\tau_{*},\bm{k}_{3})\rangle_{c}\underset{k_{3}\ll
k_{1}\sim
k_{2}}{\approx}F_{1}+F_{2}\approx-\frac{H^{6}k_{1}\tau_{*}^{4}}{4k_{3}^{3}M_{\rm
Pl}^{4}\epsilon^{2}}$
$\displaystyle\approx-\frac{d\log(k_{1}^{3}\langle{\dot{\zeta}}(\tau_{*},\bm{k}_{1}){\dot{\zeta}}(\tau_{*},-\bm{k}_{1})\rangle_{c})}{d\log
k_{1}}\langle{\zeta}(\tau_{*},\bm{k}_{3}){\zeta}(\tau_{*},-\bm{k}_{3})\rangle_{c}\langle{\dot{\zeta}}(\tau_{*},\bm{k}_{1}){\dot{\zeta}}(\tau_{*},-\bm{k}_{1})\rangle_{c}\,,$
(46)

which again corresponds to the consistency relation. Thus, we have shown that
the EOM terms are responsible for the consistency relation in the squeezed
limit.

We nevertheless emphasize that whether the bulk or the EOM terms are
responsible for the consistency relation does not really make sense. We can
change the bulk, boundary, and EOM terms according to our preference by using
the identity shown in Appendix A.

It would be worth noting that, in the operator formalism, the Heisenberg
operator $\dot{\hat{\zeta}}$ differs from that in the interaction picture
$\dot{\hat{\zeta}}^{I}$ as discussed, e.g., in Ref. [32], and accordingly, one
has to take it into account in evaluating correlation functions. This type of
difference does not appear in the path-integral case since the original source
function is formally of the Heisenberg operator, and therefore, the correction
appearing in the operator formalism does not appear. In other words, such a
correction would be automatically taken into account perturbatively. This is
the reason why naively evaluated $\langle\dot{\zeta}\dot{\zeta}\zeta\rangle$
shows the correct soft behavior without any corrections associated with the
difference between the Heisenberg and the interaction pictures.

###  III.2 Operator formalism viewpoint

We briefly discuss the consistency relation within the operator formalism. As
we have discussed, the boundary terms contribute to the computation of the
three-point correlation function, whereas the EOM terms trivially vanish in
the operator formalism of the interaction picture. Since the role of the
boundary operators was already discussed in [10, 11] for soft modes, we only
focus on the consistency relation of $\langle\zeta\zeta\zeta\rangle$ with the
first two fields in the sub-Hubble regime ($k_{1}\tau_{*}\gg 1$ and
$k_{2}\tau_{*}\gg 1$), while the other is in the super-Hubble regime
($k_{3}\tau_{*}\ll 1$). In the corresponding squeezed limit ($k_{3}\ll
k_{1}\sim k_{2}$), the leading-order term can be found from the contribution
of a single boundary operator171717This is not a Hamiltonian operator as it is
dimensionless.

$\hat{H}_{\partial}=-\int d^{3}\bm{x}\hat{\mathcal{L}}_{\partial}\supset\int
d^{3}\bm{x}\frac{M_{\rm
Pl}^{2}a\epsilon}{H}\hat{\zeta}^{\prime}\hat{\zeta}\hat{\zeta}^{\prime}\biggl{|}_{\tau=\tau_{*}}\,,$
(47)

where we have symmetrized the interaction such that the operator becomes
Hermitian. The boundary operator is located at the time where the operators
are inserted, which enables us to use the equal-time canonical commutation
relations. For $\langle\zeta\zeta\zeta\rangle$, we find

$\displaystyle\langle\hat{\zeta}(\bm{x}_{1})\hat{\zeta}(\bm{x}_{2})\hat{\zeta}(\bm{x}_{3})\rangle_{c}$
$\displaystyle\supset$ $\displaystyle\frac{{\rm i}M_{\rm
Pl}^{2}a\epsilon}{H}\int
d^{3}\bm{x}[\hat{\zeta}_{\bm{x}}^{\prime}\hat{\zeta}_{\bm{x}}\hat{\zeta}_{\bm{x}}^{\prime},\hat{\zeta}_{1}\hat{\zeta}_{2}\hat{\zeta}_{3}]$
(48) $\displaystyle=$
$\displaystyle\frac{1}{2aH}\left[\langle(\hat{\zeta}^{\prime}_{1}\hat{\zeta}_{1}+\hat{\zeta}_{1}\hat{\zeta}^{\prime}_{1})\hat{\zeta}_{2}\hat{\zeta}_{3}\rangle+\hat{\zeta}_{1}(\hat{\zeta}^{\prime}_{2}\hat{\zeta}_{2}+\hat{\zeta}_{2}\hat{\zeta}^{\prime}_{2})\hat{\zeta}_{3}+\hat{\zeta}_{1}\hat{\zeta}_{2}(\hat{\zeta}^{\prime}_{3}\hat{\zeta}_{3}+\hat{\zeta}_{3}\hat{\zeta}^{\prime}_{3})\right],$

where we have used abbreviation $\hat{\zeta}(\bm{x})=\hat{\zeta}_{\bm{x}}$,
and $\hat{\zeta}_{i}=\hat{\zeta}(\bm{x}_{i})$ and omitted the time
$\tau=\tau_{*}$. Evaluating the Fourier transform of Eq. (48) yields

$\displaystyle\langle\hat{\zeta}(\bm{k}_{1})\hat{\zeta}(\bm{k}_{2})\hat{\zeta}(\bm{k}_{3})\rangle_{c}\supset$
$\displaystyle\frac{1}{2aH}\frac{d}{d\tau}\left[|\zeta_{k_{2}}(\tau)|^{2}|\zeta_{k_{3}}(\tau)|^{2}+(\text{cyclic})\right]_{\tau\to\tau_{*}}\underset{k_{3}\ll
k_{1}\sim k_{2}}{\longrightarrow}-\frac{H^{4}k_{1}^{2}\tau_{*}^{2}}{8M_{\rm
Pl}^{4}\epsilon^{2}k_{1}^{3}k_{3}^{3}}+\mathcal{O}(k_{3}^{-1})\,,$ (49)

which coincides with the leading-order slow-roll contribution to Eq. (36) for
$k_{1}\tau_{*}\sim k_{2}\tau_{*}\gg 1$. Thus, as the EOM terms are required in
the path-integral formalism, the boundary operators are necessary to reproduce
physically important properties such as the consistency relations.

In the above calculation, the two modes are in the sub-Hubble regime, with the
other mode in the super-Hubble regime. Performing a similar calculation where
all three modes are in the super-Hubble regime, it follows that both the
boundary and the bulk terms are necessary to prove the consistency relation.

##  IV Comment on computation of higher-order terms

In this section, we give a few comments on the computation of higher-order
corrections such as loops. As is clear from our discussion so far, the
boundary (EOM) terms are not negligible for the operator (path-integral)
formalism. In particular, we are mostly concerned with the large-scale modes
generated and exiting the Hubble radius during inflation, which is expected to
be “decoupled”. As we have seen, for the consistency relation to be satisfied,
we need to correctly include the boundary and the EOM terms, even for higher-
order corrections.

In operator formalism, it is rather straightforward to take the boundary terms
into account by simply leaving it in the interaction Hamiltonian. However, we
should notice that it does not look like boundary terms at higher order. More
specifically, at second order, the expectation value of an operator
$\hat{\mathcal{O}}(\tau_{*})$ receives

$\displaystyle\langle\hat{\mathcal{O}}(\tau_{*})\rangle$
$\displaystyle\supset$ $\displaystyle{\rm
i}^{2}\int^{\tau_{*}}_{-\infty}d\tau_{1}\int^{\tau_{1}}_{-\infty}d\tau_{2}\left\langle\left[\hat{H}^{I}_{\rm
bulk}(\tau_{2})+\frac{d}{d\tau_{2}}\hat{g}^{I}(\tau_{2}),\left[\hat{H}^{I}_{\rm
bulk}(\tau_{1})+\frac{d}{d\tau_{1}}\hat{g}^{I}(\tau_{1}),\hat{\mathcal{O}}^{I}(\tau_{*})\right]\right]\right\rangle$
(50) $\displaystyle=$
$\displaystyle-\int^{\tau_{*}}_{-\infty}d\tau_{1}\int^{\tau_{1}}_{-\infty}d\tau_{2}\left\langle\left[\hat{H}_{\rm
bulk}(\tau_{2}),\left[\hat{H}_{\rm
bulk}(\tau_{1})+\frac{d}{d\tau_{1}}\hat{g}(\tau_{1}),\hat{\mathcal{O}}(\tau_{*})\right]\right]\right\rangle$
$\displaystyle-\int^{\tau_{*}}_{-\infty}d\tau_{1}\left\langle\left[\hat{g}(\tau_{1}),\left[\hat{H}_{\rm
bulk}(\tau_{1})+\frac{d}{d\tau_{1}}\hat{g}(\tau_{1}),\hat{\mathcal{O}}(\tau_{*})\right]\right]\right\rangle\,,$

where we have assumed $\hat{g}(\tau)\to 0$ as $\tau\to-\infty$ for the
equality in the second line. The point is that the total time-derivative term
inside the commutator cannot be simply performed. In the second line on the
RHS of Eq. (50), the boundary terms “behave” like a bulk term in general,
which would be understood, e.g., from the fact that the operator

$\displaystyle\hat{u}(\tau_{1})\frac{d}{d\tau_{1}}\hat{g}(\tau_{1})$ (51)

is no longer a total time derivative of some operator in general, where
$\hat{u}(\tau_{1})\equiv\int^{\tau_{1}}_{-\infty}d\tau_{2}\hat{H}_{\rm
bulk}(\tau_{2})$. Therefore, it would be practical to think of the total time
derivative operator to be a part of the bulk Hamiltonian. As long as we keep
the total time-derivative terms, we are free to throw away the EOM terms
within the interaction picture operator formalism for the reason explained
before. In this sense, despite complexities, it is rather straightforward to
compute higher-order corrections.

It might be useful to also note that the last term appearing in Eq. (50) can
be rewritten as

$\displaystyle-\int^{\tau_{*}}_{-\infty}d\tau_{1}\left\langle\left[\hat{g}^{I}(\tau_{1}),\left[\hat{H}^{I}_{\rm
int}(\tau_{1}),\hat{\mathcal{O}}^{I}(\tau_{*})\right]\right]\right\rangle=-\int^{\tau_{*}}_{-\infty}d\tau_{1}\left\langle\left[\hat{H}^{I}_{4}(\tau_{1}),\hat{\mathcal{O}}(\tau_{*})\right]-\left[\hat{H}^{I}_{\rm
int}(\tau_{1}),\left[\hat{g}^{I}(\tau_{1}),\hat{\mathcal{O}}(\tau_{*})\right]\right]\right\rangle,$
(52)

where we have defined an effective quartic interaction

$\displaystyle\hat{H}_{4}^{I}(\tau_{1})=\left[\hat{g}^{I}(\tau_{1}),\hat{H}^{I}(\tau_{1})\right]\,.$
(53)

Since this operator is made of an equal-time commutator, only the part
containing the canonical commutation relation gives spatial $\delta$-function,
with which the interactions become quartic contact terms. The appearance of
quartic terms from cubic interactions is quite similar to that in Ref. [32],
although their relation is not yet clear since the quartic terms shown in Ref.
[32] have a different origin. We will find that the appearance of effective
quartic contact terms is in common with the path-integral formalism as will be
shown below.

In the path-integral formalism, we have to deal with the EOM terms rather than
the boundary terms. As we have shown, their role is equivalent to the field
redefinition, but an advantage of keeping it as interaction terms rather than
field redefinition is that we may straightforwardly compute correlation
functions without careful treatment of the path-integral measure, which is
required when one performs field redefinition, in general. More specifically,
we often use the field redefinition of the form $\zeta\to\zeta+f(\zeta)$,
which leads to the functional Jacobian $\left|{\rm
Det}\left(\delta^{4}(x-y)+\delta f(\zeta(x))/\delta\zeta(y)\right)\right|$. Of
course, such an advantage would be a small technical issue, and in the case of
the S-matrix in flat spacetime, it is known that field redefinition does not
change the result [36, 37] and the nontrivial Jacobian factor vanishes by
working with dimensional regularization scheme [38] (except for chiral
anomalies). It is not immediately clear whether a similar argument applies to
the case of the cosmological correlators as the terms removed by field
redefinition contain $\zeta^{\prime}$ as well as nonlocal operators containing
$\partial^{-2}$. As far as we know, such an issue has not been addressed in
the literature and it is worth investigating further. We will leave it for
future work. At least, we argue that such a subtle issue does not appear as
long as we leave EOM terms since we do not need any field redefinition.

We give a few comments on higher-order corrections including the EOM terms. As
already discussed, the EOM terms contribute to correlation functions only when
the field with the EOM derivative operator is contracted by (anti-)time-
ordered propagators, which yields the spacetime $\delta$-function. For
instance, we consider the contraction of the following two interaction
vertices,

$\displaystyle+{\rm i}\int
d\tau_{1}d^{3}\bm{x}_{1}\,f\left(\zeta^{+}(\tau_{1},\bm{x}_{1})\right)\contraction{\mathcal{D}_{2}}{\zeta^{+}}{(\tau_{1},\bm{x}_{1})\times(+{\rm
i})\int
d\tau_{2}d^{3}\bm{x}_{2}\sum}{g}\mathcal{D}_{2}\zeta^{+}(\tau_{1},\bm{x}_{1})\times(+{\rm
i})\int d\tau_{2}d^{3}\bm{x}_{2}\sum g\left(\zeta^{+}(\tau_{2},\bm{x}_{2})$
$\displaystyle\supset-\prod_{i=1}^{2}\int
d\tau_{i}d^{3}\bm{x}_{i}\,\mathcal{D}_{2}G^{++}(\tau_{1},\bm{x}_{1};\tau_{2},\bm{x}_{2})f(\zeta^{+}(\tau_{1},\bm{x}_{1}))\frac{\delta
g}{\delta\zeta^{+}}(\tau_{2},\bm{x}_{2})=+{\rm i}\int
d\tau_{1}d\bm{x}_{1}f(\zeta^{+})\frac{\delta g}{\delta\zeta^{+}}\,.$ (54)

If they are originally cubic interactions, the above term is a contact four-
point interaction like the case of the boundary and the bulk contraction in
the operator formalism mentioned above. Note that, in the above expression, we
have taken just a single contraction, which is just a part of the full
contractions. Thus, the inclusion of the EOM term to the second order
effectively yields “tree level” contributions. This is simply expected from
the viewpoint of the field redefinition as well. The appearance of the quartic
contact terms is observed in the operator formalism as discussed above, and it
would be related to the quartic terms discussed in Ref. [32], which play a
crucial role in reproducing the consistency relation and proving the time-
independence of curvature perturbations.

As we have shown explicitly, with the set of interaction terms (31)-(33), the
boundary or the EOM terms are necessary to reproduce the consistency relations
correctly, and therefore, one has to include them properly, otherwise one may
find incorrect ultraviolet behavior of the loop corrections, which would be
relevant to the claim made in Refs. [25, 27].

##  V Summary

In this paper, we have discussed how to deal with the boundary and EOM terms
appearing in cosmological QFT calculations. In operator formalism, the EOM
term trivially vanishes, but the boundary terms contribute to the correlation
function in a nontrivial manner. On the other hand, in the path-integral
formalism, both of them seem to contribute to the correlation function, but
(at least) at the lowest order, the boundary terms do not contribute under our
prescription where the final time slice $\tau_{f}$ is set later than
$\tau_{*}$, the time of the operators whose expectation is evaluated.

The crucial difference between the two formalisms boils down to how
derivatives in the interactions are treated. In the operator formalism, the
derivatives directly act on operators and we evaluated the differentiated
operators in the correlation functions, whereas in the path-integral
formalism, we evaluate all by the free propagators and the derivatives act on
the propagators rather than operators. Either method would give a correct
result, so long as we carefully treat the boundary and the EOM terms. These
observations resolve some misleading statements, particularly for the EOM
terms that they trivially vanish. With the understanding developed in this
work, it would be rather straightforward to consider higher-order corrections,
for which we have also given a few comments. In particular, it has been known
that consistency relations play crucial roles in proving the conservation of
super-Hubble curvature perturbations against possible time-dependence from
loop corrections (see e.g., [32]), and as we have partially seen, the boundary
or the EOM terms may become responsible for the consistency relations of sub-
Hubble modes, which would contribute to the ultraviolet part of the loop
calculations. As long as the boundary and the EOM terms are correctly treated,
we would be able to compute arbitrarily higher-order corrections precisely.

In future work, we can apply the two formalisms to the calculation of one-loop
corrections for single-field inflation with a temporal ultra-slow-roll phase.
It was also shown that the same consistency relation as the Maldacena’s one
holds for more general slow-variation single-field inflation in Horndeski’s
theories with second-order field equations of motion [39] (see also Ref.
[40]), by picking up dominant boundary and EOM terms in the cubic action with
the interaction picture. It will be of interest to formally apply our
prescription of the operator and path-integral formalisms to such more general
theories to provide a more rigorous proof for the consistency relations.

## Note added

While completing this work, we became aware of a related work [41], in which
the authors discussed the same issue regarding the total time derivative and
the EOM terms in great detail. We mostly agree with their results, but our
proposal regarding the boundary terms within the path-integral formalism seems
rather simple as our argument shows that the boundary terms can be neglected
within it by causality at any order in the perturbation series (unless we
consider more involved time contours such as an out-of-time-ordered contour).
Therefore, it is not so cumbersome to use the known path-integral formalism as
one only needs to take the EOM terms. In our opinion, it is not yet clear
whether the newly proposed canonical transformation approach in Ref. [41] is
more practical than the standard path-integral approach leaving the EOM terms
since the latter does not cause possible quantum anomalies that may be
associated with the canonical transformation.

## Acknowledgements

We thank Tomohiro Fujita, Mohammad Ali Gorji, Jason Kristiano, and Yuichiro
Tada for useful discussions. ST is supported by the Grant-in-Aid for
Scientific Research Fund of the JSPS No. 22K03642 and Waseda University
Special Research Project No. 2023C-473. YY is supported by Waseda University
Grant for Special Research Projects (Project number: 2023C-584).

##  Appendix A Ambiguity of the boundary and the EOM terms

The boundary and the EOM terms can be arbitrarily changed by the following
procedure: One finds the following identity

$\displaystyle 0=\int d^{4}x\left[{\mathcal{D}}_{2}\zeta
f(\zeta)-\frac{d}{dt}\left(2a^{3}\epsilon\dot{\zeta}f(\zeta)\right)+2a^{3}\epsilon\dot{\zeta}\dot{f}(\zeta)-2a\epsilon\,\partial\zeta\partial
f(\zeta)\right]\,,$ (55)

where we have neglected the total spatial derivative terms. Adding such a
“zero” to the action changes the bulk, boundary, and EOM terms. Such ambiguity
can be used to change the interaction terms. As we have shown in the main
text, any of them cannot be neglected in general.

##  Appendix B Remarks on boundary terms

We discuss some subtleties of the total time-derivative term in the
Hamiltonian. As an illustration, we use the following simple time-dependent
Hamiltonian in quantum mechanics:

$\displaystyle\hat{H}[\hat{\phi}(t),\hat{\pi}(t);t]=\hat{H}_{0}[\hat{\phi}(t),\hat{\pi}(t);t]+\hat{A}[\hat{\phi}(t),\hat{\pi}(t);t]+\frac{d}{dt}\hat{B}[\hat{\phi}(t),\hat{\pi}(t);t]\,,$
(56)

where $\hat{\pi}$ is the canonical momentum operator of $\hat{\phi}$,
$\hat{H}_{0}[\hat{\phi}(t),\hat{\pi}(t);t]$ is the free part of the
Hamiltonian, and $\hat{A}[\hat{\phi}(t),\hat{\pi}(t);t]$ is the (bulk)
interaction term. We have assumed that the Hamiltonian has explicit time-
dependence, e.g., in the couplings as the cosmological models. The above
definition of the Hamiltonian already has a subtlety: The Hamiltonian contains
the total time derivative of $\hat{B}$, but its definition depends on the
Hamiltonian itself since

$\displaystyle\frac{d}{dt}\hat{B}[\hat{\phi}(t),\hat{\pi}(t);t]=\hat{U}^{-1}_{S}(t,t_{0})\frac{\partial}{\partial
t}\hat{B}[\hat{\phi}(t_{0}),\hat{\pi}(t_{0});t]\hat{U}_{S}(t,t_{0})+{\rm
i}\left[\hat{H}[\hat{\phi}(t),\hat{\pi}(t);t],\hat{B}[\hat{\phi}(t),\hat{\pi}(t);t]\right]\,,$
(57)

where $\hat{U}_{S}(t,t_{0})$ is the Schrödinger picture time-evolution
operator satisfying

$\displaystyle{\rm
i}\frac{d}{dt}\hat{U}_{S}(t,t_{0})=\hat{H}[\hat{\phi}(t_{0}),\hat{\pi}(t_{0});t]\hat{U}_{S}(t,t_{0})\,,$
(58)

where $t_{0}$ denotes the initial time. In defining the Schrödinger picture
Hamiltonian $\hat{H}[\hat{\phi}(t_{0}),\hat{\pi}(t_{0});t]\equiv\hat{h}(t)$,
the consistency with the Heisenberg picture requires that

$\displaystyle\hat{h}(t)=\hat{h}_{0}(t)+\hat{A}_{0}(t)+\frac{\partial}{\partial
t}\hat{B}_{0}(t)+{\rm i}[\hat{h}(t),\hat{B}_{0}(t)]\,,$ (59)

where $\hat{h}_{0}(t)=\hat{H}_{0}[\hat{\phi}(t_{0}),\hat{\pi}(t_{0});t]$,
$\hat{A}_{0}(t)=\hat{A}[\hat{\phi}(t_{0}),\hat{\pi}(t_{0});t]$, and
$\hat{B}_{0}(t)=\hat{B}[\hat{\phi}(t_{0}),\hat{\pi}(t_{0});t]$. We note that
the last term is necessary to reproduce the total time derivative term in the
original Hamiltonian in the Heisenberg picture
$\hat{H}=\hat{U}_{S}^{-1}\hat{h}\hat{U}_{S}$. One may use the interaction
picture based on the free Hamiltonian $\hat{H}_{0}$, but one would immediately
find that the interaction Hamiltonian is yet dependent on the full interaction
Hamiltonian. Thus, the time evolution of either states or operators must be in
a self-consistent way. As far as we know, there is no way to define a
Hamiltonian without a term containing the Hamiltonian itself.

The above issue appears not only in the Hamiltonian with a boundary term but
the system with terms containing time-derivatives on the canonical variables
$\phi(t),\pi(t)$. This is somewhat reasonable as higher time derivatives lead
to ghost poles in general, which gives rise to additional degrees of freedom
in the theory. We expect that such higher derivative terms do not appear in
ultraviolet complete theories but only in their effective field theories. In
particular, there are no terms containing higher-derivative terms in general
relativity with minimal matter couplings, and there is a Hamiltonian that has
no subtleties discussed above, and one has to define a theory with it.

Suppose that our Hamiltonian does not have the total time derivative terms
posing the above issue. In this case, one can first quantize the system and
move to the interaction picture, and then we may integrate the interaction
Hamiltonian by part, which effectively induces the total time derivative terms
as is the case of metric perturbations in general relativity. Nevertheless, it
should be noted that, even if higher-derivative terms are absent, it is
nontrivial to find a full Lagrangian if the Lagrangian contains powers of
$\dot{\phi}$ since its canonical conjugate is no longer
$\pi_{\phi}\propto\dot{\phi}$. In such a case, one has to carefully define the
Hamiltonian.

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# Precision Higgs Measurements at the 250 GeV ILC

The Graduate University for Advanced Studies (SOKENDAI) and High Energy
Accelerator Research Organization (KEK)
E-mail:

on behalf of Linear Collider Collaboration (LCC) Physics Working Group

###### Abstract:

The plan for the International Linear Collider (ILC) is now being prepared as
a staged design, with the first stage of 2 ${\rm ab^{-1}}$ at 250 GeV and
later stages achieving the full project specifications with 4 ${\rm ab^{-1}}$
at 500 GeV. The talk presents the capabilities for precision Higgs boson
measurements at 250 GeV. It is shown that the 250 GeV stage of the ILC will
already provide many compelling results in Higgs physics, with new
measurements unavailable at the Large Hadron Collider, model-independent
determinations of key parameters, and possible discrimination of a variety of
scenarios for new physics.

The talk is based on Phys. Rev. D97 (2018) 053003 and arXiv:1710.07621.

##  1 Introduction

The 125 GeV Higgs boson was discovered at the Large Hadron Collider (LHC) in
2012 [1, 2, 3]. Measurements on Higgs couplings and properties with LHC data
of approximately 100 ${\rm fb^{-1}}$ , which yield precisions in the 10%–20%
range, indicate that the new particle is consistent with the Higgs boson in
the Standard Model (SM). Meanwhile, the 13 TeV LHC run has not provided any
indications of a new particle. This indicates that additional Higgs bosons or
supersymmetric particles, which appear in theories beyond the SM (BSM) such as
minimal supersymmetric SM (MSSM) where the Higgs sector is extended as various
two Higgs doublet models (2HDMs), will be considerably heavy.

In such a case, the theories such as the 2HDMs phenomenologically indicate
implications that coupling strengths of the lightest BSM Higgs boson to the SM
particles will be identical to those of the SM Higgs boson due to suppression
depending on the scale of the heaver Higgs boson, which is known as the
decoupling limit of the Higgs models [4]. When generally adapting the
decoupling limit for the BSM models in the same manner, typical deviations of
the Higgs couplings from the SM expectations due to corrections by the
theoretical parameters are predicted to be at a 10% level or rather smaller
[5]. Therefore, to detect indications of the deviations and its pattern,
reaching precision of 1% level is required, and the International Linear
Collider (ILC) with the 250 GeV operation is expected to be capable of
measuring the Higgs couplings with the 1% precision.

##  2 A framework based on effective field theory

Traditionally the $\kappa$-framework has been often employed to evaluate the
Higgs couplings [6]. An application of the $\kappa$-framework, however, is
limited only for the leading order consideration because of problems on
internal consistency and model-independence confronting the evaluation:

  * •

There is no relation between the scaling parameters of $\kappa_{V}$ although
symmetry breaking is imposed with the assumption of $SU(2)\times U(1)$ gauge
symmetry.

  * •

All contributions due to higher-order radiative corrections by the SM
particles are regarded as one single $\kappa$ factor although it is composed
of several of the other $\kappa$ factors.

  * •

Assuming new (next to the leading order) tensor structures composed of field
strength tensors, which have momentum dependence, certain production processes
and the corresponding decay processes do not match each other in terms of the
scaling.

In contrast, a framework of the Effective Field Theory (EFT) can deal with
those problems naturally by adding proper physics-driven higher order field
operators. The effective Lagrangian under the ILC-EFT framework has been
established based on dimension-6 field operators, where the number of
parameters which must be decided simultaneously is 23 [7]: the EFT
coefficients, renormalizing SM parameters, and Higgs decays unpredicted in the
SM.

Advantages of the EFT are not only to deal in the problems but be possible to
increase sensitive observables which can contribute to improvement of the
Higgs couplings since the EFT can make the parameters relate. Even observables
which are not directly related to the Higgs boson become useful to increase
the precision. Furthermore, by exploiting beam polarization of the ILC, which
is one of the strong functions, the number of sensitive observables can be
doubled. Therefore, all the potential of the ILC is possible to be extracted
under the EFT framework, within which a percent precision can be reached for
the Higgs couplings.

##  3 Higgs couplings measurements at 250 GeV

The following observables and information are available to determine the Higgs
couplings:

  * •

Direct observations of the Higgs related observables such as inclusive
measurement of the $ZZH$ coupling based on the recoil mass technique, various
$\sigma\times\rm{BR}$ measurements [5], and also differential cross-section
measurements of the Higgs related processes, which are sensitive to new
Lorentz structures [8].

  * •

Observations on the triple gauge couplings through the $e^{+}e^{-}\rightarrow
W^{+}W^{-}$ process [9, 10].

  * •

Electroweak precision observables for constraining the renormalized SM
parameters [9].

  * •

Combinations of the beam polarizations can double observables derived from
difference of reactions because of the polarization asymmetry.

  * •

Higgs observables on $\gamma Z$ and $\gamma\gamma$ provided by the High
Luminosity LHC (HL-LHC) operation [11]. The ratios of $\rm{BR}_{\gamma
Z}/\rm{BR}_{\gamma\gamma}$ and $\rm{BR}_{ZZ}/\rm{BR}_{\gamma\gamma}$ are
employed to remove systematics of the measurements.

Figure 1: The plot is referred from the reference [12], which illustrates the
precisions of the Higgs couplings evaluated with the global fitting under the
ILC-EFT framework (light green). The precisions achievable with ATLAS under
the $\kappa$-framework [11] is also shown as the comparison (red).

Using these observables mentioned above, global fitting was performed to
evaluate the reachable precisions of the Higgs couplings to the SM particles
at the 250 GeV ILC, which is described in detail in two papers [12, 13].
Figure. 1 is illustrating the precisions of the Higgs couplings given by ATLAS
at HL-LHC, in the $\kappa$-framework and after combination with the 250 GeV
ILC under the EFT-framework. The histogram that includes the 250 GeV ILC data
indicates that the precisions of the couplings for $ZZ$ and $WW$ reach 1%
below and ones for $b\bar{b}$, $\tau\tau$, and $gg$, which are restricted due
to systematics at the LHC, are significantly improved. It is also shown that
synergy between HL-LHC and ILC enables improvement in $\gamma\gamma$ and
$\gamma Z$ measurements. The precision for $c\bar{c}$ can be achieved with the
250 GeV ILC because of huge QCD backgrounds in the LHC although one for
$t\bar{t}$ is not improved due to the given energy. The precisions of the
couplings for the vector bosons, up-type and down-type quarks, and third-
generation lepton will reach the 1% level.

##  4 Conclusion

We illustrated that the precisions for the Higgs couplings to the SM particles
achievable with the 250 GeV ILC is the 1% level or below, corresponding to the
precisions for discriminating the various BSM models that can not be covered
by the HL-LHC. The capability of 250 GeV ILC to discriminate the BSM models
are discussed in the reference [13] by exemplifying the representative BSM
models. The above illustrates that the ILC has a capability to indicate models
and a scale of eventual new physics.

## References

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