Large gauge transformation, soft theorem, and Infrared divergence in inflationary spacetime
HJE
Large gauge transformation, soft theorem, and Infrared divergence in inflationary spacetime
Takahiro Tanaka 0 1 3
Yuko Urakawa 0 1 2
0 Chikusa , Nagoya 4648602 , Japan
1 Sakyo , Kyoto 6068502 , Japan
2 Department of Physics and Astrophysics, Nagoya University
3 Department of Physics, Kyoto University
It is widely known that the primordial curvature perturbation ζ has several universal properties in the infrared (IR) such as the soft theorem, which is also known as the consistency relation, and the conservation in time. They are valid in rather general single clock models of inflation. It has been argued that these universal properties are deeply related to the large gauge transformations in inflationary spacetime. However, the invariance under the large gauge transformations is not sufficient to show these IR properties. In this paper, we show that the locality condition is crucial to show the consistency relation and the conservation of ζ. This argument also can apply to an interacting system with the inflaton and heavy fields which have arbitrary integer spins, including higher spin fields, which may be motivated from string theory. We will also show that the locality condition guarantees the cancellation of the IR divergences in a certain class of variables whose correlation functions resemble cosmologically observable quantities.
Effective Field Theories; Cosmology of Theories beyond the SM; Gauge

Symmetry
1 Introduction
2 Asymptotic symmetry and soft theorem
2.1
Large gauge transformations
2.2
2.3
3.1
3.2
3.3
2.1.1
2.1.2
2.3.1
2.3.2
Dilatation as a large gauge transformation
Two different prescriptions of dilatation
Dilatation invariance and Noether charge
Revisiting consistency relation
Condition for consistency relation
Conditions on the homogeneous mode and hard modes
3
Massive particles with arbitrary spins
Setup of the problem
Soft theorem for heavy fields with nonzero spins
Effective action
3.3.1
3.3.2
Influence functional
Soft theorem and effective action
3.4 Conservation of ζ
4 IR divergences of inflationary correlators
4.1
Overview of IR divergence problem
4.2 Cancellation of IR divergence and the locality condition
5 Relevance and irrelevance of soft graviton insertion
6 Concluding remarks
A Perturbative and nonperturbative definition of Noether charge B Computing the effective action 1 3
deeply related to the large gauge transformations. The large gauge transformation is
– 1 –
a local symmetry transformation which does not approach the unity at the infinity of
spacetime [1, 2].
The large gauge transformations which play an important role in cosmology are those
defined at a time constant slicing and diverge at the spatial infinity. In single field models
of inflation, the invariance under the dilatation and shear transformations, which are both
large gauge transformations, directly ensures the massless property of the curvature
perturbation ζ and the gravitational waves γij . It is widely known that these massless fields ζ
and γij have several universal properties in the IR, which are valid in rather general single
field models of inflation. First, at the treelevel computation, ζ and γij are nonlinearly
conserved in time in the IR limit [3–11]. (The conservation of ζ was discussed by including
the radiative corrections of ζ in refs. [12, 13].) Second, the influence from the soft modes of
ζ and γij on the hard modes are, rather generically, characterized by the socalled
consistency relation [14, 15], which is an example of the soft theorem in cosmology. It has been
suggested, e.g., in refs. [3, 16–20], that these universal properties are both consequences
of the invariance under the large gauge transformations. However, it was also revealed
that this invariance is not sufficient to derive the consistency relation. In refs. [18–20], the
analyticity in the soft limit was additionally imposed to derive the consistency relation.
The consistency relation has been understood as a relation between the npoint
correlation function with n hard modes and the (n + 1)point correlation function with n hard
modes and one soft mode. In deriving the consistency relation, usually, the validity of
the perturbation theory is presumed. However, this is not trivially guaranteed, because
the massless fields ζ and γij are not screened in the large scale limit and their radiative
corrections can diverge due to their unsuppressed IR contributions. Therefore, unless the
IR divergence is regularized without violating the dilatation invariance, the consistency
relation cannot be welldefined as a relation between the correlation functions. Notice that
introducing a naive IR cutoff which regularizes the IR radiative corrections can violate the
dilatation invariance, which is a crucial property for the consistency relation.
The goal of this paper is to clarify the relation between the large gauge transformations
and the abovementioned universal properties of ζ and γij in the IR, i.e., the soft theorem
(also known as the consistency relation), the conservation in time, and the possible
appearance of the IR divergence. In particular, along the line of the argument in refs. [18–20],
we will clarify the physical meaning of the analyticity in the soft limit, which is needed to
derive the consistency relation. In ref. [18], it was argued that this condition is related to
the locality of theory. In this paper, we scrutinize this argument, clarifying what we need
to require as the locality condition, because the locality has a broad meaning. We also
discuss what the consistency relation actually describes, taking into account the possibility
that the perturbative prediction can be spoiled by the IR divergence.
We will also show that the large gauge transformations play a crucial role also in
discussing the radiative corrections from massive fields. Being motivated by string theory,
which may predict the presence of higher spin fields in the four dimensional theory
obtained after compactification, we consider massive fields with arbitrary integer spins. After
deriving the consistency relation for the hard modes of the massive fields with nonzero
spins, we discuss the condition that the curvature perturbation ζ stops evolving in time in
– 2 –
the soft limit under the influence of the radiative corrections from the massive fields. This
is a generalization of our previous study [19], where we considered the radiative corrections
from a massive scalar field, to a massive field with a general integer spin. As was argued in
ref. [21], exploring an imprint of the higher spin fields in the primordial nonGaussianity of
ζ may provide a unique probe of string theory. (See also refs. [22–25].) Our study provides
the conditions for the absence of such an imprint made after the Hubble crossing time of
the comoving scale of our interest during inflation.
Recently, the relation among the large gauge transformations, the soft theorem, and
the IR divergence has been studied intensively for gauge theories in an asymptotically flat
spacetime [26–28]. (For a review, see ref. [29].) In ref. [26], it was shown that Weinberg’s
soft theorem [30], which describes the influence of the soft photon and graviton, can be
obtained as a WardTakahashi identity of the asymptotic symmetries at the null infinity.
(The soft theorem for massless higher spin fields in an asymptotically flat spacetime was
discussed in ref. [31].) More recently, (the cancellation of) the IR divergence was discussed,
using the Noether charge of the asymptotic symmetries [32]. At first sight, the IR
structures for the gauge fields in the asymptotically flat spacetime have similar properties to
those for the primordial perturbations ζ and γij in cosmology. We give a closer look at
this apparent similarity.
This paper is organized as follows. In section 2, considering single field models of
inflation, we discuss the relation between the large gauge transformation and the consistency
relation for ζ. Here, we seek for a deeper understanding about the locality condition as
the necessary condition to derive the consistency relation for ζ. In section 3, we will show
that the discussion in section 2 can be straightforwardly extended to an interacting system
composed of the inflaton and the massive fields with nonzero spins. We also show that the
locality condition implies both the consistency relation for the hard modes of the massive
fields and the conservation of ζ. In section 4, we show that the locality condition also
ensures the cancellation of the IR divergence for a certain class of variables. In section 5, we
briefly show that the discussion for the soft graviton proceeds almost in parallel to the one
for the soft modes of ζ, discussed in section 2–4. In section 6, after summarizing our results,
we discuss a similarity and an apparent difference between the IR properties for the gauge
fields in the asymptotically flat spacetime and those for the primordial perturbations.
2
Asymptotic symmetry and soft theorem
In this section, we discuss the relation between the large gauge transformations and the
soft theorem in cosmology, also known as the consistency relation. We clarify the condition
that derives the consistency relation and discuss its physical meaning. Before we start our
discussion, we clarify what we mean by “soft” and “hard.” The soft modes mean the modes
with k/aH → 0 but k 6= 0, and the hard modes mean the remaining inhomogeneous modes,
including the super Hubble modes with k/aH < 1 but excluding the limit k/aH → 0. By
contrast, we describe the wave vector of a longer mode and that of a shorter mode as kL and
∼
kS, respectively, simply based on the ratio between these wavenumbers, i.e., kL/kS ≪ 1.
– 3 –
HJEP10(27)
As we will discuss in section 4, in the limit k/aH ≪ 1, a perturbative expansion can break
down in computing some quantity and taking this limit requires a careful treatment.
the soft modes of ζ and γij in an inflationary spacetime. In line with refs. [1, 2], we define the
large gauge transformation as follows. A local symmetry denotes a symmetry under a
transformation which is parametrized by a spacetime dependent function, while a global
symmetry denotes a symmetry under a transformation by a spacetime independent function.
identity at the infinity and the latter does not. In refs. [26, 29], it was shown that the
soft theorem for the photons and the gravitons in the asymptotically flat spacetime can be
derived from the WardTakahashi identities for large gauge transformations which do not
vanish on J ±.
line element:
2.1.1
Dilatation as a large gauge transformation
First, let us clarify the prescription we adopt. In this paper, we use the ADM form of the
ds2 = −N 2dt2 + hij (dxi + N idt)(dxj + N j dt) ,
where we introduced the lapse function N , the shift vector N i, and the spatial metric hij .
We determine the time slicing, employing the uniform field gauge:
We express the spatial metric hij as
where γij is set to traceless. As spatial gauge conditions, we impose
To discuss the soft modes of the primordial perturbations in the spatially flat FRW
background, we consider the large gauge transformations, which do not vanish at the spatial
infinity on a time constant surface. This large gauge transformation was first discussed
in the context of cosmology by Weinberg in ref. [3]. In the unitary gauge, where the
fluctuation of the inflaton vanishes, we consider, in particular, the dilatation:
where s is a constant parameter. Under the dilatation, the curvature perturbation ζ
transforms as
ζ(t, x) → ζs(t, x) = ζ(t, e−sx) − s .
δφ = 0 .
hij = a2e2ζ [eγ ]ij ,
∂iγij = 0 .
xi → esxi ,
– 4 –
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
The change of ζ is given by
Δsζ(t, x) = −s(1 + x · ∂xζ(t, x)) + O(s2) .
(2.7)
The classical action in a diffeomorphism (Diff) invariant theory remains invariant under
the transformation of ζ given in eq. (2.6). As one may expect from the fact that the
dilatation shifts ζ by −s, the dilatation invariance is related to the massless property of ζ,
which implies that ζ is conserved at large scales in single clock inflation.
2.1.2
Two different prescriptions of dilatation
The dilatation invariance may be somehow confusing, because it also appears as a part
of the de Sitter invariance by changing the time coordinate simultaneously. The Killing
vector which corresponds to this transformation is given by
−η∂η − xi∂i ,
where η denotes the conformal time. The dilatation symmetry in de Sitter group states
that the time shift can be compensated by the scale transformation. Since inflation has
to end at some point, the time translation symmetry needs to be broken in the context of
inflationary scenario.
The time translation symmetry is broken, when the physical frequency ωph becomes
well below Λb with Λ4
b ≡ φ˙2. In the effective field theory of inflation [33], the Goldstone
mode, the pion π, is introduced to restore the invariance under the time reparametrization
t → t + ξ ,
π → π − ξ
(2.8)
in the symmetry breaking phase. With this construction, the pion Lagrangian nonlinearly
preserves the invariance since π appears only in the combination t + π. Through the
coupling with the metric perturbations, the pion acquires the mass of mπ = O(√ε1H).
The relation between the dilatation discussed in section 2.1.1 and the one discussed
here is somewhat puzzling. As we will discuss in the following section, the former is
preserved in an arbitrary quasi FRW spacetime, while the latter is a part of the de Sitter
symmetry and is broken below the symmetry breaking scale Λb. Related to this point,
preserving the former dilatation invariance directly ensures that ζ should be massless. On
the other hand, there is no simple argument which shows the massless property of ζ in the
latter prescription, where ζ is related to the Goldstone mode π as ζ = −Hπ. Related to
this point, recall that the pion acquires the mass mπ through the coupling with the metric
perturbation for ωph <∼ mπ.
1
value, the pion is no longer massless. The curvature perturbation is sometimes said to be
a Goldstone mode, since π = −ζ/H is the Goldstone mode. This statement can cause
a confusion, because it may sound as if ζ preserves the shift symmetry, being massless,
because it is a Goldstone boson associated with the breaking of the de Sitter symmetry.
Therefore, in the regime where ζ approaches a constant
1Recall that the Goldstone mode for a global symmetry is not necessarily massless in a Lorentz violating
background.
– 5 –
In the following, by the dilatation, we mean the former one, which is a spatial
coordinate transformation without the time coordinate change. Considering this dilatation, we
discuss the IR behaviour of ζ such as the consistency relation and the conservation in time.
We will emphasize that the invariance under this dilatation, which is a part of the large
gauge transformations, is preserved as well in the quantized system and therefore there is
no spontaneous symmetry breaking in this prescription.
2.2
Dilatation invariance and Noether charge
In this subsection, we discuss several implications of the dilatation invariance in single
clock inflation. Following ref. [16], we define the Noether charge for the dilatation as
d3x [Δsζ(t, x)πζ (t, x) + πζ (t, x)Δsζ(t, x)] ,
where πζ denotes the conjugate momentum of ζ, which satisfies
[ζ(t, x), πζ (t, y)] = iδ(x − y) .
[Qζ , H] = 0 ,
[Qζ , ζ(x)] = −iΔsζ(x) .
– 6 –
Since the Hamiltonian for ζ is invariant under the dilatation, we obtain
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
which implies that Qζ is independent of time. The Noether charge is a generator of the
dilatation transformation and satisfies
Using the Fourier components of the fields2 we can rewrite the Noether charge Qζ as
s Z
d3k
Qζ = −sπζ,k=0 − 2
(2π)3 {ζk, k · ∂kπζ, −k} + O(s2) .
In performing the Fourier transformation, we did not drop the surface term. Therefore,
the charge Qζ given in eq. (2.9) is identical to the one given in eq. (2.13). The first term
of Qζ only operates on the k = 0 mode. The Noether charge Qζ can diverge due to the
IR modes, because it is an integral over the infinite spatial volume. In the following, we
neglect higher order terms of O(s2).
Equation (2.13) gives the nonperturbative definition of the Noether charge for the
dilatation. Since an explicit computation usually relies on perturbative expansion in the
interaction picture, one may want to introduce the Noether charge by using the fields in
2We use the convention of the Fourier transformation:
f (x) =
(2π)3
Z d3k eik·xfˆ(k) ,
fˆ(k) =
Z d3xe−ik·xf (x) .
Here, the commutation relation for the Fourier modes of ζ and πζ is given by [ζk, πζ k′ ] = i(2π)3δ(k + k′).
the interaction picture as3
where ΔsζI (t, x) is given by
I
d3x ΔsζI (t, x)πζI (t, x) + πζI (t, x)ΔsζI (t, x) ,
ΔsζI (t, x) = −s(1 + x · ∂xζI (t, x)) .
(2.15)
(2.16)
When we perform the dilatation transformation in the interaction picture or after the
perturbative transformation, there is one caveat which should be kept in mind. Changing
order of performing the finite dilatation transformation and performing the perturbative
ζ
expansion leads us to a different answer. In order words, performing the dilatation in the
interaction picture (with a use of QI ) and performing the dilatation in the Heisenberg
picture (with Qζ ) are different, while this discrepancy disappears, when we consider the
infinitesimal transformation generated by dQζ /dss→0. A lesson from here is that for a
legitimate prescription, the dilatation transformation should be performed in the
nonperturbative Heisenberg picture. A more detailed discussion can be found in appendix A.
2.3
Revisiting consistency relation
In this subsection, we discuss the condition(s) to derive the consistency relation of ζ. We
will see that while the dilatation invariance plays a crucial role, it is neither sufficient nor
even necessary to derive the consistency relation.
2.3.1
Condition for consistency relation
The consistency relation for ζ was first derived by Maldacena from an explicit computation
of the bispectrum of ζ [14]. Afterwards, the consistency relation was derived in a rather
general setup of single field models of inflation [15], while several examples that do not
satisfy the consistency relation were also reported [34–36] (see also refs. [37, 38]).
In ref. [14], it was argued that the bispectrum in the squeezed limit, which describes
the correlation between the long mode kL and short mode kS, can be computed by
considering the influence of the long mode on the short mode. This influence can be described
as the dilatation kS → e−ζkL kS. While ζ is not invariant under the dilatation
transformation,4 we can construct another variable which is invariant under the dilatation
transformation [39, 40]. In ref. [41], we showed that the leading contributions to the bispectrum
of such an invariant variable in the squeezed limit are canceled, choosing the adiabatic
vacuum. In this computation, the squeezed bispectrum of ζ satisfies Maldacena’s consistency
relation, which is the key to ensure the cancellation.
3Notice that since the free Hamiltonian H0 is not invariant under the dilatation, i.e.,
hH0, QIζi 6= 0 ,
(2.14)
in time.
large gauge transformations.
in contrast to Qζ, which is time independent, the “charge” defined in the interaction picture QIζ varies
4The curvature perturbation ζ is invariant under the small gauge transformations but not under the
– 7 –
The consistency relation does not hold for an arbitrary quantum state even in single
field models. In refs. [17, 18, 20], it has been directly and indirectly suggested that the
dilatation invariance of the quantum state:
Qζ Ψi = 0
(2.17)
is crucial to derive the consistency relation. In ref. [17], it was shown that the invariance
of the 1PI effective action for ζ under the dilatation leads to a set of identities which relate
the npoint function and the (n + 1)point function for ζ. What we directly obtain from
these identities, which are the so called WardTakahashi (WT) identity, is “the consistency
relation,” which relates the npoint function with n hard modes to the (n+1)point function
with the homogeneous mode ζk=0 in addition to the n hard modes. The only difference from
the consistency relation is in that the inserted mode is not the soft mode ζk with k 6= 0.
Along this line, in refs. [20, 42, 43], “the consistency relation” was derived starting
with the dilatation invariance of the wave function in the ζ representation, Ψ[ζ] ≡ hζΨi,
where  ζi is the normalized eigenstate of ζ, i.e.,
Ψ[ζs(t, x)] = Ψ[ζ(t, x)] ,
(2.18)
which directly follows from the invariance of the quantum state under the dilatation (2.17).5
If and only if the WT identity which describes the insertion of ζk=0 can be extended to
the relation which describes the insertion of the soft mode ζk with k 6= 0, we obtain the
consistency relation:
lim C
kn→0
(n)({ki}n)
P (kn)
!
= −
X ki · ∂ki + 3(n − 2) C
(n−1)({ki}n−1) ,
(2.19)
where C(n) denotes the npoint function of ζ with the momentum conservation factor
n−1
i=2
(2π)3δ
n
X ki
i=1
!
removed. Since we additionally need to impose that this extension is possible, requesting
the dilatation invariance of the quantum state (2.17) is not enough to derive the consistency
relation.
Now, the question is “What is the physical meaning of the additional condition that
allows the WT identity to be smoothly extended to the consistency relation, which describes
the insertion of the soft mode?” This issue was first addressed in ref. [18], where it was
argued that this condition is related to the locality of the theory. Even if the original
theory is local, the Lagrangian density for ζ in the unitary gauge becomes nonlocal due to
5For instance, at O(s), eq. (2.18) gives
0 =
Z d3x Δsζ(t, x)
δ
δζ(t, x) ψ[ζ] ,
which can be obtained from 0 = hζ(x)Qζψi by inserting the expression of the Noether charge, given in
eq. (2.9). Here, we used hζπζψi ∝ δψ[ζ]/δζ.
– 8 –
the presence of the Lagrange multipliers, the lapse function and the shift vector, which are
given by solving the elliptic equations. For instance, at the linear order in perturbation,
the shift vector Ni includes a contribution given by
∂iNi ⊃ εζ˙ ,
which introduces nonlocal interaction vertices. In the standard slowroll inflation, ζ˙k is
suppressed in the limit k → 0 as ζ˙k = O(kpζk) with p ≥ 1. However, in the absence of this
suppression, the coefficients of the nonlocal interaction vertices in the Fourier space can
be singular in the limit k → 0. Then, Ni should be determined discontinuously at k = 0
in order to avoid the singular behaviour.
Along this line, our purpose of this section is to sharpen the relation between the
condition for the locality and the condition of being able to extend the “consistency
relation” with the insertion of the homogeneous mode to the consistency relation with the
soft mode k 6= 0, clarifying the physical meaning of the condition. As was mentioned
above, the Lagrangian density for ζ (and also for the gravitational waves) is nonlocal in
the sense that the Lagrangian density cannot be solely determined by the dynamical fields
at each spacetime point. For a classical theory, the condition for the smooth extension
of the homogeneous mode k = 0 to the soft modes k 6= 0 with a suitable falloff at the
spatial infinity is nothing but the one to pick up the Weinberg’s adiabatic mode [3]. We
elaborate the physical meaning of this condition for a quantum theory by using the Noether
charge Qζ . As will be discussed in the next section, with the use of the Noether charge, a
generalization to the case with higher spin fields proceeds straightforwardly.
2.3.2
Conditions on the homogeneous mode and hard modes
As was mentioned in section 2.3.1, the validity of the consistency relation is deeply related
to the invariance of the wave function or the effective action under the dilatation, which is
a large gauge transformation. Maldacena derived the consistency relation for the treelevel
bispectrum in the squeezed limit, i.e., (2.19) with n = 3, choosing the adiabatic vacuum
(or the Euclidean vacuum) [14]. This vacuum also can be defined nonperturbatively by
requesting the regularity of correlation functions in the limits t → −∞(1 ± iǫ).6 Here, the
time path is rotated towards the imaginary axis in the distant past. This serves one of
the examples of the quantum state Ψi, which preserves the dilatation invariance, since
this definition does not artificially introduce any specific scale.
At perturbative level,
a correlation function for the Euclidean vacuum can be calculated by adopting the iǫ
prescription. Here, the (free) mode function should be chosen to be the one for the adiabatic
vacuum (a.k.a, the BunchDavies vacuum in de Sitter limit) [44].
In order to study the condition for the dilatation invariance of the quantum state (2.17),
we decompose the wave function in terms of the eigenstates of the spatial average of ζ all
over a time constant slicing, e.g., at the end of inflation,
momenta. Therefore, all the momenta go to the UV in these limits.
¯
ζ ≡
R d3xζ(x)
R d3x
,
– 9 –
(2.20)
where ζ¯c is a cnumber eigenvalue. In order to distinguish the eigenvalues of ζ¯, which are
cnumbers, from the operator ζ¯, we put the index c on the eigenvalues. In eq. (2.21), we
factorized the wave function of ζ¯, ψ(ζ¯c), from hζ¯c Ψi, while absorbing the phase into  Ψiζ¯c , as
From the normalization condition for  Ψiζ¯c , the amplitude of the wave function ψ(ζ¯c)
is unambiguously defined. Since the quantum state  Ψi also includes the inhomogeneous
modes with k 6= 0, hζ¯c Ψi should be understood as a vector in infinite dimensional Hilbert
space. We express the normalized quantum state for all the modes with k 6= 0 obtained
by the projection of  Ψi to the eigen state  ζ¯ci as  Ψiζ¯c . Using eq. (2.12), we obtain
Using this condition, we can express the dilatation invariance condition for the
quan0 =
Z
dζ¯c
−s ∂ψ(ζ¯c)
∂ζ¯c
  ζ¯ci Ψiζ¯c + ψ(ζ¯c)  ζ¯ci iQζ − s ∂ζ¯c  Ψiζ¯c .
∂
Operating hζ¯c′  on eq. (2.27), the real and imaginary parts, respectively, give
The eigenstate  ζ¯c i satisfies
Ψi =
Z
dζ¯c ψ(ζ¯c)  ζ¯c i Ψiζ¯c .
ζ¯ ζ¯c i = ζ¯c ζ¯c i ,

The first condition (2.28) requires that the amplitude of the wave function ψ(ζ¯c), which
represents the probability distribution of ζ¯, should be flat in the direction of ζ¯ in the Hilbert
space. The condition (2.28) is satisfied, e.g., for a Gaussian wave function whose variance
Since an operation of eiQζ shifts the eigenvalue of ζ¯ by s, i.e.,
we obtain
tum state  Ψi, (2.17), as
and
iQζ , ζ¯ = −s .
eiQζ  ζ¯ci =  ζ¯c + si ,
iQζ  ζ¯ci = s ∂ζ¯c  ζ¯ci .
∂
∂
∂ζ¯c ψ(ζ¯c) = 0 ,
iQζ  Ψiζ¯c = s ∂ζ¯c  Ψiζ¯c .
∂
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
blows up in the limit of the homogeneous mode, which is the case for a nearly scale
invariant power spectrum. While the wave function for ζ¯ becomes nonnormalizable, as will be
discussed in section 4, this does not create any problems as far as we compute a certain set
of quantities which mimic to observable quantities. The condition (2.28) requires that the
probability distribution should be nonperturbatively flat in the direction of the
homogeneous mode ζ¯. The second condition may be a little more nontrivial. The quantum state of
the inhomogeneous modes generically changes under a variation of the homogeneous mode
ζ¯. The condition (2.29) restricts how the quantum state of the inhomogeneous modes should
respond to the change of the homogeneous mode, which is expressed by the operation of Qζ.
Since the consistency relation describes an insertion of an inhomogeneous mode with
k 6= 0 that suitably falls off at the spatial infinity, we need to extend the above argument to
inhomogeneous modes. For this purpose, we introduce the generator of a spatial dependent
dilatation given by
QζW (x) ≡ 2
1 Z d3x′W (x′ − x) Δsζ(t, x′)πζ(t, x′) + πζ(t, x′)Δsζ(t, x′) ,
(2.30)
induces the dilatation only for the fields at x′ with x−x  ∼
where W (x) denotes a smooth window function which is normalized as R d3x W (x) = 1
and which vanishes at x >∼ L. Here, we set L to be of order of 1/kL. The generator QζW (x)
′ < L. The Noether charge (2.9) is
defined by the integral all over the time constant slicing and hence the integral does not
converge in general. The introduction of the window function can make the integral converge.
Since the domain of integration is finite, the generator QζW (x) depends on where we choose
the center of the integral domain, x. Performing the Fourier transformation, we obtain
QζW (kL) = −s−kL πkL +
1 Z d3k
2
(2π)3 {ζk, k · ∂kπkL−k} + O(s2) ,
(2.31)
where Wˆ (k) denotes the Fourier mode of the window function, which is normalized as
and vanishes for k ≫ 1/L. Here, we introduced
We focus on the field within a large volume of O(Lc3). We set Lc to be much larger than
all the wavelengths, i.e., Lc ≫ 1/kL, 1/kS, and we will send Lc → ∞ after our computation.
Next, we introduce a smeared field in momentum space defined by
k→0
lim Wˆ (k) = 1 ,
sk ≡ sWˆ (k) .
ζkL ≡ Lc3 Z d3k′ Wˆ (kL − k′) ζk′ ,
˜
3
i=1
Wˆ (k) = Y θ (2Lc)−1− ki .
(2.32)
(2.33)
(2.34)
with
˜
ζkL describes a collective mode with the representative wavenumber kL and when we
evaluate ζ˜ in the position space by performing the inverse Fourier transformation, it decays
outside the local volume of O(Lc3). The fluctuations outside the local volume can be
described as the modes which are orthogonal to the collective mode ζ˜kL. Since the correlations
between the fluctuations for x ≪ Lc and those outside the local volume of O(Lc3) are
negligibly small, in what follows, we neglect the fluctuations outside the local volume, which
will disappear after taking the limit Lc → ∞.
The commutation relation of QζW with the long mode ζkL is given by
hiQζW (kL) , ζ˜pLi = −(2πLc)3s−kLWˆ (kL + pL) .
(2.35)
Equation (2.35) states that the generator QζW (kL) shifts the collective soft mode ζ˜kL by
−skL(2πLc)3. We will find that the factor (2πLc)3, which blows up in the limit Lc → ∞,
is cancelled out in the final expression of the consistency relation as it should be. Whilst,
the commutation relation with the short mode ζpS is given by
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)
iQζW (kL) , ζpS ≃ s−kL ∂pS (pS ζpS+kL) ,
where we approximated kS + kL as kS.
Repeating the argument around eqs. (2.21)–(2.23) except that the homogeneous mode
ζ¯ is now replaced with a collective inhomogeneous soft modes ζ˜kL, we expand the quantum
state  Ψi in terms of the orthonormal basis { ζ˜pcLi}, which are the eigenstates of ζ˜pL, as
Ψi =
Z dζ˜pcL ψ(ζ˜pcL)  ζ˜pcL i Ψiζ˜pcL ,
where we factorized the amplitude of the wave function, ψ(ζ˜kcL), from hζ˜kcLΨi, as before.
Using eq. (2.35), we obtain
iQζW (kL) ζ˜−ckL i = (2πLc)3s−kL ∂ζ˜−ckL
˜c
 ζ−kL i .
As we have already discussed, the dilatation invariance requires the conditions (2.28)
and (2.29). In particular, the second condition (2.29) restricts how the inhomogeneous
modes respond to the insertion of the homogeneous mode ζ¯. In the following, we will show
that when this condition can be extended to the soft mode with kL 6= 0, i.e.,
∂
∂
iQζW (kL) Ψiζ˜−ckL
= (2πLc)3s−kL ∂ζ˜−ckL
 Ψiζ˜−ckL
is fulfilled, we can derive the consistency relation for ζ as shown below.
In order to show the consistency relation by using the condition (2.39), we evaluate
hΨ  [iQζW (kL), ζkS1 · · · ζkSn]  Ψi
in two ways: first by operating iQζW (kL) on the quantum state  Ψi and second by
considering the change of the short wavelength modes ζkS under the inhomogeneous dilatation,
where Pζ (kL) denotes the power spectrum of ζ. This is because the variance of ζ˜kcL is
given by
h  ζ˜kL 2i = (2π)3Lc6 Z d3kWˆ (kL − k)Wˆ (−kL + k)Pζ (kL) ≃ (2πLc)3Pζ (kL) ,
(2.43)
independent variable from ζ˜−ckL, since ζ˜kcL = ζ˜−c∗kL holds from the reality of ζ.
where the last equality is exact in the limit Lc → ∞. Remember that ζ˜kcL is not an
Using eq. (2.42), we obtain
ψ(ζ˜−ckL) ∝ exp
ζ˜kcLζ−kL
˜c
− 4(2πLc)3Pζ (kL)
!
,
∂
∂ζ˜−ckL
˜c
ζkL
s.l.
ψ(ζ˜−ckL) ≈ − 2(2πLc)3Pζ (kL) ψ(ζ˜−ckL) .
expressed by the commutation relation with iQζW (kL). When the condition (2.39) is
satisfied, using eq. (2.38), we obtain
iQζW (kL) Ψi = (2πLc)3s−kL
∂
Z dζ˜−ckL ψ(ζ˜−ckL) ∂ζ˜−ckL
˜c
 ζ−kL i Ψiζ˜−ckL
= −(2πLc)3s−kL
Z dζ˜−ckL
∂ζ˜−ckL
∂ψ(ζ˜−ckL)  ζ˜−ckL i Ψiζ˜−ckL
.
When we neglect the nonlinear contributions of the soft modes,7 the wave function of
ζ˜kcL is given by the Gaussian distribution function. Since the square of ψ(ζ˜kcL) gives the
probability distribution, we can express the amplitude of the Gaussian wave function as
operator ζ˜kL, which commutes with the integral over ζ˜kcL, we obtain
Here and hereafter, we use s≈.l. to express that we approximate the wave function of the soft
mode by the above Gaussian distribution function. Replacing the eigenvalue ζ˜kcL with the
iQζW (kL) Ψi s≈.l. 2Ps−ζ(kkLL) ζ˜kL Ψi .
Using this expression, we arrive at
hΨ [iQζW (kL), ζkS1 · · · ζkSn] Ψi s≈.l. − Pζ (kL)
s−kL hΨ ζ˜kLζkS1 · · · ζkSn Ψi .
Meanwhile, using eq. (2.36), we obtain
hΨ [iQζW (kL), ζkS1 · · · ζkSn] Ψi = s−kL X ∂kSikSihΨ ζkS1 · · · ζkSn Ψi .
7Here, all the interaction vertexes connected to more than two soft modes are neglected.
n
i=1
Equating these two expressions derived from the two different ways, we have
−hΨ ζ˜kL ζkS1 · · · ζkSn  Ψi ≈ Pζ (kL) X ∂kSi kSihΨ ζkS1 · · · ζkSn  Ψi .
s.l.
s.l.
n
The ordinary form of the consistency relation (2.19) is the one obtained by removing the
delta function, which describes the momentum conservation, from the expression
−hΨ ζpL ζkS1 · · · ζkSn  Ψi ≈ Pζ (pL) X ∂kSi kSihΨ ζkS1 · · · ζkSn  Ψi .
If we take the average of eq. (2.49) operating Lc3 R d3pLWˆ (kL − pL), we recover eq. (2.48).
After we take the limit Lc → ∞, which is automatically required when we take the soft
limit kL → 0, the averaging window becomes infinitesimally narrow. Therefore, we can
conclude that eq. (2.48) is equivalent to the ordinary consistency relation.
Under the Gaussian approximation of  ψ(ζ˜kcL ) , the only assumption imposed to derive
the consistency relation (2.19) is eq. (2.39), which states that the influence of
inhomogeneous dilatation is identical to shifting the collective soft mode ζ˜kL which interacts with the
hard modes by +(2πLc)3skL . (The factor (2πLc)3 here is an artifact caused by discussing
the Fourier space collective mode.) The consistency relation can be obtained, when the
condition (2.29), which was required to preserve the dilatation invariance of the quantum
state, can be extended to the inhomogeneous soft mode with kL 6= 0. In this sense, the
condition (2.39) can be understood as a quantum version of the condition for Weinberg’s
adiabatic mode.
The condition (2.39) cannot be satisfied, in case the (linear) soft mode ζkL does not
stop evolving in time in the limit kL → 0 as in models with multi light fields and also in
nonattractor models.8 In fact, the condition (2.39) states that performing the inhomogeneous
dilatation transformation that induces the timeindependent shift of ζkL , is equivalent to
shifting (the eigen value of) ζkL at the evaluation time, which implies that the inserted soft
mode should be dominated by a constant contribution.
Now, we show that the condition (2.39) can be indeed understood as the locality
Using the complete set of  ζLc i, we decompose the quantum state  Ψ i as
condition. For this purpose, we consider a set of eigenstates for the Fourier mode ζkL
which satisfies ζkL  ζL i = ζkL  ζL i instead of the eigenstates for the collective mode ζ˜kL .
c c c
 Ψ i =
Z
DζL ψ(ζLc )  ζL i  Ψ iζLc ,
c
(2.48)
(2.49)
HJEP10(27)
(2.50)
(2.51)
where DζL represents the functional integral over all soft modes. The extension of eq. (2.39)
to the case of continuous modes will be
iQζW (kL) ΨiζLc = sWˆ (−kL) δζ−ckL
 ΨiζLc .
δ
8According to our understanding, in solid inflation [
45
], the anisotropic pressure plays the role of an
additional degree of freedom, leading to the nonconservation of ζ and also the violation of the consistency
relation. This is consistent with the statement shown in ref. [46] that the consistency relation can be
recovered by taking angular average over the long mode.
Using this relation, we obtain
iQζW (x) Ψiζ˜Lc = s
Z d3kL e−ikL·x Wˆ (kL) δζkL  Ψ iζLc = s
δ
(2π)3
δ
δζL(x)  Ψ iζLc ,
where
ζL(x) ≡
Z (d23πk)L3 Wˆ (kL)eikL·xζkL
is the coarse grained field corresponding to the degrees of freedom of soft modes. Let
us imagine a set of the separate universes whose sizes are of O(L). The operator QζW (x)
induces the dilatation only within the separate universe centered at x. The condition (2.52)
states that the impact of the soft mode on the quantum state of short modes is limited
only to the influence which is equivalent to the inhomogeneous dilatation in the separate
universe.
Notice that since (the amplitude of) the wave function is not flat in the direction of
the inhomogeneous mode of ζ, i.e., ∂ψ(ζ˜kL )/∂ζ˜kL 6= 0 , as seen in eq. (2.44), the quantum
state  Ψi does not remain invariant under the inhomogeneous dilatation QζW (kL), i.e.,
QζW (kL) Ψi 6= 0 .
As shown in eq. (2.45), operating the generator of the inhomogeneous dilatation inserts the
soft mode ζ˜kL with kL 6= 0, which changes the quantum state, even if the state is invariant
under the homogeneous dilatation. As will be discussed in section 4, because of that,
choosing a quantum state which is invariant under the dilatation is not enough to guarantee
the IR regularity. Let us emphasize that there is no spontaneous symmetry breaking for
the large gauge transformations: the symmetry under the dilatation, generated by Qζ , is
preserved also after the quantization, while the inhomogeneous dilatation, generated by
QζW , is not a symmetry of the classical action.
Although we assumed that the wave function of the soft mode, ψ(ζ˜kcL ), is given by
the Gaussian distribution, the nonlinear interactions of the hard modes are fully kept.
Therefore, the consistency relation thus derived can apply to a much more general setup
compared to the original one by Maldacena in ref. [14]. For instance, the wavelengths of the
hard modes can be arbitrary as far as kSL ≃ kS/kL ≫ 1, i.e., they are not necessarily in
super Hubble scales. In addition, obviously the same argument as above can apply, even if
the time coordinates of the short modes ζkSi (ti) with i = 1, · · · , n are different among them.
Because of the Gaussian approximation for ψ(ζ˜kcL ), the consistency relation derived
here do not contain nonlinear interactions of the soft modes. Notice that since there is no
approximation for the soft modes in the locality condition (2.39) or (2.52), when we include
the nonlinear contributions of the soft modes, the same condition leads to the consistency
relation.9 However, the nonlinear interactions of the soft modes in general yield the IR
divergences through their radiative corrections. Therefore, this extension requires a more
careful consideration. This issue will be discussed in section 4.
9When we include the nonlinear contributions of the soft modes, the dilatation also changes the
argument of ζ and the transformation of ζkL under the dilatation is not the simple shift. In order not to change
the argument, we need to introduce the window function in a physical distance such as the geodesic distance.
(2.52)
(2.53)
(2.54)
In this section, we consider an influence of heavy fields with arbitrary spins, which interact
with the inflaton directly or indirectly through the gravitational interaction. The argument
in this section is a generalization of the one in ref. [19], which showed the consistency
relation for a heavy scalar field and the conservation of ζ, taking into account radiative
corrections of the heavy scalar field. In the previous section, we derived the condition
for the consistency relation, using the generator(s) of the inhomogeneous dilatation. This
argument can be extended to the cases with the radiative corrections of massive nonzero
spin fields straightforwardly. In this section and in appendix B, to keep generality, we
consider a (d + 1)dimensional spacetime.
In this section, we consider a heavy field whose mass is MS and spin is S ≥ 0, including
higher spin fields. We only consider the mass range where there is no instability [21, 47],
e.g., M2 ≥ 2H for a spin 2 field. For our purpose, we do not need to specify the detail
of the interaction for the interacting system with the inflaton and the massive fields. We
simply express the action as
with
S[δg, χ] = Sad[δg] + Sχ[δg, χ] ,
dt ddx ad edζ(x)f {iα}(φ, δg)O{iα}(x) ,
where χ and O{iα} denote a set of heavy fields χ
I with I = 1, 2, · · · and a composite
operator of χ, respectively, and δg denotes the set of the metric perturbations, N , Ni, ζ,
and γij . Here, the action Sad[δg] only includes the metric perturbations in the unitary
gauge defined by the condition δφ = 0 and it is identical to the action in single field models
of inflation.
The heavy fields and the inflaton φ also can interact directly. For the present purpose,
we do not need to specify the composite operators O{iα}. We only need to specify their
scaling dimensions Δα, i.e., they transform as
O{iα}(t, xs) = e−ΔαsO{iα}(t, x)
s
under the dilatation transformation x → xs = esx, where iα denotes tensor indices. While
the heavy fields can be a fermion with a half integer spin, we assume that δg interacts with
χ only through the composite operators which transform as tensors (with an integer spin)
under coordinate transformations. A composite operator with n tensor (lower) indices has a
scaling dimension n, e.g., Δ = 0 for a scalar composite operator and Δ = 1 for a vector one.
3.2
Soft theorem for heavy fields with nonzero spins
In this subsection, we derive the consistency relation or the soft theorem in the presence
of the heavy fields, extending the discussion in the previous section. In the previous
section, expanding the quantum state  Ψi as in eq. (2.21), we derived the conditions (2.28)
(3.1)
(3.2)
(3.3)
and (2.29) by requesting the invariance of the quantum state under the dilatation. Here,
repeating the same argument except that now the quantum state  Ψi also includes the
heavy fields in addition to the inhomogeneous modes of ζ, we obtain the same conditions
as eqs. (2.28) and (2.29) from the dilatation invariance of  Ψi. Since  Ψiζ¯c also includes
the degrees of freedom for the heavy fields, the condition (2.29) restricts how both of the
inhomogeneous modes of ζ and the heavy fields should respond to the dilatation
transformation. Similarly, when the condition (2.29) can be extrapolated to the inhomogeneous
dilatation with the falloff at the spatial infinity by replacing the homogeneous mode ζ¯c
in (2.29) with the soft mode ζ˜kL, i.e., when the locality condition (2.39) is fulfilled also
in the interacting system with the inflaton and the heavy fields, we can derive the soft
theorem, which describe the influence of the soft mode.
Recall that the consistency relation can be derived by evaluating the change of the
quantum state  Ψi and the change of the operators for the heavy fields. In order to derive
the soft theorem for the heavy fields, we consider
hΨ [iQζW (kL), O{iα1}kS1(t1) · · · O{iαn}kSn(tn)] Ψi ,
where O{iα}kS (t) denotes the Fourier mode of O{iα}(x) with kL/kS ≪ 1. Repeating the
same argument and taking the limit Lc → ∞, we find that the condition (2.39) implies
eq. (2.45). Then, using eq. (2.45), we can compute the change of the quantum state under
the inhomogeneous dilatation as
hΨ [iQζW (kL), O{iα1}kS1(t1) · · · O{iαn}kSn(tn)] Ψi
s.l.
≈ − Pζ(kL)
s−kL hΨ ζkLO{iα1}kS1(t1) · · · O{iαn}kSn(tn) Ψi .
We equate this expression with the one obtained by computing the change of the composite
operators O{iα}k under the (inhomogeneous) dilatation. Since O{iα}k transforms as in
eq. (3.3), we obtain
hiQζW (kL), O{iα1}kS i = s−kL
kS − Δα O{iα1}kS .
∂
∂kS
Likewise in the discussion for the single field case, whether the dilatation parameter is
homogeneous or inhomogeneous does not affect the transformation of the short modes.
Using this expression, we obtain
hΨ [iQζW (kL), O{iα1}kS1(t1) · · · O{iαn}kSn(tn)] Ψi
n
i=1
s.l.
≈ s−kL X (∂kSikSi − Δαi) hΨ O{iα1}kS1(t1) · · · O{iαn}kSn(tn) Ψi .
Equating these two expressions and sending Lc to the infinity, we obtain the consistency
relation for the heavy fields as
(3.4)
(3.5)
(3.6)
(3.7)
kL→0
s.l.
≈ −
n
i=2
X kSi · ∂kSi
∂
lim hΨζkLO{iα1}kS1(t1) · · · O{iαn}kSn(tn)Ψi′
Pζ(kL)
+d(n−1)−Δ hΨO{iα1}kS1(t1) · · · O{iαn}kSn(tn)Ψi′,
(3.8)
!
where we defined Δ ≡ Pin=1 Δαi . We put a prime to denote the correlation functions
without the multiplicative factor (2π)d and the delta function which expresses the momentum
conservation.
In order to show that the radiative corrections of the heavy fields do not induce any
time evolution of ζ at large scales when the condition (2.39) is satisfied, we compute the
effective action for ζ by integrating out the heavy fields χ in the closed time path (or
the inin) formalism. In particular, the contributions of the heavy fields χ are described
by the Feynman and Vernon’s influence functional [48, 49]. In this subsection, we briefly
summarize the way to calculate the influence functional and the effective action. We will
see that now the argument to show the conservation proceeds almost in parallel to the one
for the heavy scalar field, discussed in ref. [19].
vature perturbation ζ is given by
hΨ T ζ(x1) · · · ζ(xn) Ψi
Performing the path integral along the closed time path, the npoint function of the
cur=
R Dδg+dy R Dχ+ R Dδg−dy R Dχ− ζ+(x1) · · · ζ+(xn) eiS[δg+, χ+]−iS[δg−, χ−]
R Dδg+dy R Dχ+ R Dδg−dy R Dχ− eiS[δg+, χ+]−iS[δg−, χ−]
,
(3.9)
where we double the fields: δg+ and χ+ denote the fields defined along the path from the
past infinity to the time t and δg− and χ− denote the fields integrated from the time t
to the past infinity. Since N and Ni are the Lagrange multiplies, which are eliminated by
solving the constraint equations, we perform the path integral only regarding the dynamical
degrees of freedom δgdy
≡ (ζ, γij ) and χ. An insertion of δg+(x) into the path integral in
the numerator as above gives a correlation function in the time ordering, expressed by T ,
while an insertion of δg−(x) gives a correlator in the antitime ordering, expressed by T¯.
Separating the part which describes the radiative corrections of the heavy fields as
iSeff [δg+, δg−] ≡ ln
Z
Dχ+
Z
Dχ− eiS[δg+, χ+]−iS[δg−, χ−] ,
(3.10)
we can express the npoint function for ζ superficially as if there are only the metric
perturbations and the inflaton as
hΨ T ζ(x1) · · · ζ(xn) Ψi =
The effective action is recast into
R Dδg+dy R Dδg−dy ζ+(x1) · · · ζ+(xn) eiSeff [δg+, δg−]
R Dδg+dy R Dδg−dy eiSeff [δg+, δg−]
.
(3.11)
where Se′ff is the socalled influence functional, given by
Seff [δg+, δg−] = Sad[δg+] − Sad[δg−] + Se′ff [δg+, δg−] ,
iSe′ff [δg+, δg−] ≡ ln
Z
Dχ+
Z
Dχ− eiSχ[δg+, χ+]−iSχ[δg−, χ−] ,
(3.12)
(3.13)
where we factorized Sad[δg±] which commutes with the path integral over χ±. Here, we
only consider the correlation functions for ζ, but the effective action Seff [δg+, δg−] describes
the evolution of both ζ and γij affected by the quantum fluctuations of the heavy fields χ.
For our later use, we introduce the correlation functions of χ computed in the absence of
the metric perturbations as
.
Expanding Se′ff in terms of the metric perturbations δg = (δN, Ni, ζ, γij ), we obtain
(3.14)
(3.15)
(3.16)
(3.17)
∞
n=0
iSe′ff [δg+, δg−] ≡
X iSe′ff(n)[δg+, δg−] ,
where Se′ff(n) denotes the terms which include n δgαs, given by
iSe′ff(n)[δg+, δg−] =
1
X
n! a1=±
· · ·
X
an=±
Z
dd+1x1 · · ·
Z
dd+1xn
× δga1 (x1) · · · δgan (xn) Wδ(gna)1 ···δgan (x1, · · · , xn) ,
with the nonlocal interaction vertices induced by the heavy fields:
Wδ(gna)1 ···δgan (x1, · · · , xn) ≡ δga1 (x1) · · · δgan (xn) δg±=0
.
δniSe′ff [δg+, δg−]
In eq. (3.16), each δgam with m = 1, · · · , n should add up all the metric perturbations
δNam , Ni,am , ζam , γij am . Here and hereafter, for notational brevity, we omit the summation
symbol over δg unless necessary. Inserting eq. (3.13) into eq. (3.17), we can express the
nonlocal interaction vertices by using the correlators for χ. These expressions are summarized
in appendix B. Once we expand the effective action as in eq. (3.16), the shift symmetry
is lost at each order in perturbation about δg. In the following, we will show that we can
rewrite the effective action in such a way that the shift symmetry is manifestly preserved
by using the consistency relation.
3.3.2
Soft theorem and effective action
In order to show the conservation of ζ in the presence of the radiative corrections of the
heavy fields, here we rewrite the consistency relation (3.8). As was mentioned in the
previous section, the time coordinates of the hard modes can be different among different
composite operators. Taking an appropriate ordering of the composite operators, i.e., we
can put the index ai = ± on each composite operator O{iαi }kSi in the consistency
relation (3.8). In the following, we use the prescription introduced in the previous subsection
(see also appendix B). In particular, all the correlation functions should be understood as
being computed in the path ordering of the closed time path with the distinction of ±.
Employing the Gaussian approximation for the soft mode of ζ again, we can compute
the correlation function in the first line of eq. (3.8) as
hΨζkL O{ai1α1 }kS1 (t1) · · · O{ainαn }kSn (tn)Ψi
s.l.
≈ −i hΨ, 0ζ Sχint − ζkL O{ai1α1 }kS1 (t1) · · · O{ainαn }kSn (tn) Ψ, 0ζ i
+ i hΨ, 0ζ ζkL O{ai1α1 }kS1 (t1) · · · O{ainαn }kSn (tn) Sχint + Ψ, 0ζ i ,
where  Ψ, 0ζ i denotes the quantum state  Ψi with ζ being in noninteracting vacuum.
Here, Sχint denotes a set of the interaction vertexes which include only one ζ without
derivative and the massive fields χ and is given by
with
Sχint =
Z
dt ddx ζ(x)
δSχ
δζ(x) ζ=0
=
Z
dt
Z
ddk
δSχ
(2π)d ζk δζ ζ=0
(t, −k)
δSχ
δζ ζ=0
(t, k) ≡
Z
ddx e−ik·x δSχ
δζ(x) ζ=0
.
+
+
+
+
+
(3.18)
(3.19)
(3.20)
We express the interaction action Sχint with the heavy fields on the paths ± as Sχint ±,
respectively. Factoring out the power spectrum of ζ, we obtain
hΨζkL O{ai1α1 }kS1 (t1) · · · O{ainαn }kSn (tn)Ψi
s.l.
≈ −i
+ i
Z
Z
dt
dt
*
*
Pζ (kL)
Ψ, 0ζ
δSχ−
δζ ζ=0
(t, kL) O{ai1α1 }kS1 (t1) · · · O{ainαn }kSn (tn) Ψ, 0ζ
Ψ, 0ζ O{ai1α1 }kS1 (t1) · · · O{ainαn }kSn (tn)
(t, kL) Ψ, 0ζ
,
(3.21)
δSχ+
δζ ζ=0
where the power spectrum of ζ is canceled between the numerator and the denominator.
In the first line of eq. (3.21), we omitted the time coordinate of the soft mode ζkL , since
it should be constant in time to satisfy the condition (2.39). (If one wants to specify the
time coordinate of ζkL , we can place it along the closed time path at the end of inflation
where ζk+L = ζ
k−L .) Expressing the composite operators in the position space, we obtain
n
i=1
s.l.
≈ −i
+ i
Z
Z
dt
dt
*
*
*
δSχ−
δζ ζ=0
X{∂xi xi − (d − Δi)}
Ψ, 0ζ O{ai1α1 }(x1) · · · O{ainαn }(xn) Ψ, 0ζ
Ψ, 0ζ
(t, kL) O{ai1α1 }(x1) · · · O{ainαn }(xn) Ψ, 0ζ
Ψ, 0ζ O{ai1α1 }(x1) · · · O{ainαn }(xn)
(t, kL) Ψ, 0ζ
.
(3.22)
δSχ+
δζ ζ=0
Since we neglected the higher order contributions of the soft modes using the approximation
of s≈.l., we replaced the quantum state  Ψi in the first line with  Ψ, 0ζ i.
3.4
For the purpose of showing the conservation of ζ, let us introduce
δgˆ(x) = {N (x), e−ζ(x)Ni(x), ζ˙(x), e−ζ(x)∂xζ(x), e−ζ(x)∂x, e−2ζ(x)γij (x)} .
Since the metric perturbations transform as
Ns(t, xs) = N (t, x) ,
ζs(t, xs) + s = ζ(t, x) ,
esNi, s(t, xs) = Ni(t, x) ,
e2sγij s(t, xs) = γij (t, , x)
under the dilatation transformation, we can easily see that δgˆ(x) transform as a scalar
under the dilatation.
Taking into account that the composite operator O{iα} transforms as given in eq. (3.3),
we also can construct a scalar operator for O{iα} as e−Δαζ(x)O{iα}(x). Here, introducing
fˆ{iα}(φ, δgˆ) ≡ eΔαζ f {iα}(φ, δg)
for each f {iα} in the action (3.2), we rewrite the action for the heavy fields as
dt ddx adedζ(x)fˆ{iα}(φ, δgˆ)e−Δαζ(x)O{iα}(x) .
Since fˆ{ia} transforms as a scalar under the dilatation, the metric perturbations included
in fˆ{ia} can be expressed only in terms of δgˆ(x), which also transform as a scalar.
Taking the first and the second variations of Sχ with respect to ζ, we obtain
α
α
α
δζ2(x) δg=0
= X ad(t)(d − Δα)f {iα}O{iα}(x) ,
= X ad(t)(d − Δα)2f {iα}O{iα}(x) .
Since we expressed the action as in eq. (3.27), the terms in which the derivative operates on
fˆ{iα} vanish. Variations with respect to the other metric perturbations, δN, e−ζ Ni, e−2ζ γij ,
which are in the combination of δgˆ, yield
δSχ
δgˆ(x) δg=0
δ2Sχ
δgˆ(x)δζ(x) δg=0
= X ad(t)
∂fˆ{iα}
∂gˆ(x) δg=0
O{iα}(x) ,
∂fˆ{iα}
= X ad(t)(d − Δα) ∂gˆ(x) δg=0
α
O{iα}(x) ,
and so on. Multiplying ad(t)∂fˆ{iα}/∂g(x)δg=0 on eq. (3.22) with n = 1 and all the
remaining metric perturbations set to 0 and taking summation over α, we obtain
∂x nx Wδ(g1±) (x)o s.l. Z
≈
dty
Z
ddye−ikL·y nWδ(g2±)ζ+ (x, y) + Wδ(g2±)ζ− (x, y)o
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
HJEP10(27)
for δg = ζ, δN, Ni, γij . Notice that the derivative with respect to e−ζ Ni and the one with
respect to Ni give the same answer after setting δg to 0. The same story also follows for
γij . Here, we expressed δSχ/δζζ=0(t, kL) in eq. (3.22) in the position space. In deriving
eq. (3.22), we used the explicit forms of Wδ(g2a)1 δg˜a2
Multiplying δg± on eq. (3.32) and integrating over xμ, we finally obtain the key formula to
(x1, x2), summarized in appendix B.
show the presence of the constant solution as
Z
dd+1x {x · ∂xδg±(x)}Wδ(g1±) (x)
Z
dd+1x Z
dd+1y e−ikL·yδg±(x) nWδ(g2±)ζ+ (x, y) + Wδ(g2±)ζ− (x, y)o s.l.
≈ 0 ,
(3.33)
where we performed integration by parts. Meanwhile, multiplying e−ik′L·x on eq. (3.32)
and integrating over xμ, we obtain
Z
dd+1xe−ik′L·x Z
dd+1ye−ikL·y nWδ(g2±)ζ± (x, y) + Wδ(g2±)ζ∓ (x, y)o = 0 .
(3.34)
By adding the left hand side of eq. (3.33) multiplied by a constant parameter −s and
eq. (3.34) with δg± = ζ± multiplied by −s2/2, the linear and the quadratic terms in the
effective action can be given by
iSe′ff(1)[δg+, δg−] + iSe′ff(2)[δg+, δg−]
= X Z
a=±
dd+1x δga, s(x)Wδ(g1a)(x)
+
1
X
2! a1,a2=±
Z
+ O(δg3, δg2s, δgs2, s3) ,
dd+1x1
Z dd+1x2 δga1, s(x1)δg˜a2, s(x2)Wδga1 δg˜a2 (x1, x2)
(2)
where δgs denote the metric perturbations δg after the inhomogeneous dilatation. For
notational brevity, here we used the same notation as those for the global dilatation, given
in eqs. (3.25) and (3.24). Here, each δgi,a (i = 1, 2) sums over δNa,s, Ni,a,s, ζa,s, and γij,a,s.
In deriving eq. (3.35), we used
Wδ(g2a)1 δg˜a2 (x1, x2) = Wδg˜a2 δga1 (x2, x1) .
(2)
The first term in eq. (3.33) changes the argument of the metric perturbations in the linear
term of the metric perturbations in eq. (3.35).
We also changed the arguments of the
quadratic terms, taking into account that the modification appears only in higher orders
of perturbation. The tadpole contributions, which are the terms in the second line of
eq. (3.35) should vanish by using the background equation of motion. (See ref. [19] for a
discussion about the heavy scalar field.)
′
Equation (3.35) shows that with the use of the consistency relation, δgα(x) in Seff
can be replaced with δgα, s(x). Since the rest of the effective action, Sad, is simply the
classical action for the single field model, it also should be invariant under this replacement.
(3.35)
(3.36)
Therefore, when the locality condition (2.39) holds, the total effective action Seff preserves
the invariance under the inhomogeneous dilatation with the suitable falloff, i.e., under the
change of δgα to δgs,α. Since eqs. (3.33) and (3.34) also hold for the soft modes not only
for the homogeneous mode with k = 0, we can shift ζ(x) by an inhomogeneous but
timeindependent function instead of the homogeneous constant parameter. The invariance of
the effective action, which also includes the radiative corrections from the nonzero spin
massive fields, under the shift of the soft mode ζkL directly implies the existence of the
constant solution for ζ.10
We assumed the locality condition (2.39) to derive the consistency relation, which was
used to show the invariance under the replacement of ζ with ζs in the effective action. As
was argued in section 2.3.2, the validity of eq. (2.39) requires the linear perturbation of the
soft mode ζkL to be time independent. For that, the decaying mode of ζkL should die off
sufficiently fast after the Hubble crossing. This happens when the background trajectory
is on an attractor, e.g., when the heavy fields do not alter this nature of the background
classical trajectory. As far as the radiative corrections of the heavy fields χ do not turn the
“decaying” mode of ζkL into a growing mode, the existing constant solution should be the
dominant solution of ζ in the large scale limit. This should be the case, when the radiative
corrections of the heavy fields remain perturbative. In ref. [19], we listed the condition for
the conservation of ζ in the presence of the radiative corrections of a massive (scalar) field as
• The radiative corrections of the heavy field are perturbatively small.
• The background trajectory is on an attractor.
• The quantum system preserves the dilatation invariance.
Now the second and third conditions are rephrased by the single condition:
• The locality condition (2.39) is satisfied.
This guarantees the presence of the constant solution also for the soft modes ζkL with
kL 6= 0, not only for the homogeneous mode. The current argument also can apply to
massive fields with arbitrary spins, including higher spin fields.
4
IR divergences of inflationary correlators
In the previous two sections, we showed that the locality condition (2.39) leads to the
consistency relation and the conservation of ζ. In this section, as another related subject,
we show that the condition (2.39) also plays an important role for the cancellation of the
IR divergent contributions.
10The lapse function and the shift vector, included in the effective action (3.35), can be eliminated by
solving the Hamiltonian and momentum constraint equations and expressing them in terms of ζs as in the
single field model [14]. Since the constraint equations for δgs are simply given by replacing δg with δgs
in the constraint equations for δg, the effective action obtained after eliminating these Lagrange multiplies
obviously preserves the invariance under the replacement of ζ with ζs.
4.1
It is widely known that the loop corrections of a massless perturbation mode such as the
curvature perturbation ζ can yield various IR enhancements. (See, e.g., refs. [50–58].) In
this section, following ref. [59], we briefly summarize the IR enhancements. When we
perform the perturbative expansion in terms of the interaction picture field ζI , an interaction
vertex which includes ζI without derivative can yield the radiative correction which is
proportional to hζI2i. The super Hubble modes with k < aH contribute to hζI2i as R0aH d3k/k3,
yielding the logarithmic enhancement. We distinguish the divergent contribution due to
∼
the modes 0 ≤ k ≤ kc (IRdiv) from the convergent but secularly growing one (∝ ln a) due
to the modes kc ≤ k ≤ aH (IRsec). Here, kc denotes an IR cutoff. The IRsec, which
cannot be removed by introducing the comoving IR cutoff, originates from the accumulation
of the super Hubble modes. The accumulation of the super Hubble modes enhances the
time integral at each interaction vertex (SG), introducing another secularly growing term
proportional to ln a or increasing the power of ln a included in the integrand. These IR
enhancements also can be introduced by the soft graviton.
Cancellation of IR divergence and the locality condition
In the series of papers [39, 40, 44, 59–64], we showed that the IR enhancements, i.e.,
IRdiv, IRsec, and SG, are due to the influences from the outside of the observable region.
What we observe in cosmological measurements corresponds to a quantity evaluated in a
completely fixed gauge. Fixing the gauge conditions eliminates the influence of the gauge
degrees of freedom. However, it is not possible to determine the gauge condition outside the
observable region, even if we completely specify the way of observation. Therefore, unless
the causality is manifestly ensured as in the harmonic gauge, the degrees of freedom outside
the observable region can affect the boundary conditions of the observable region. As was
argued in refs. [39, 40], changing the boundary conditions corresponds to changing the
spatial coordinates in the local observable region. Such spatial coordinate transformations
are the large gauge transformations. We dubbed a variable which is independent of those
boundary conditions of the local observable region as a genuine gauge invariant variable.
In refs. [39, 40, 44, 59, 60, 63, 64], we showed that all the IR enhancements are cancelled
out, when we calculate a correlation function of a genuine gauge invariant variable for a
specific initial state such as the adiabatic vacuum. In ref. [63], we argued that whether
the IR enhancements in the correlation function of the genuine gauge invariant operator
disappear or not depends on the choice of the initial states. In this subsection, scrutinizing
the condition on the quantum state to ensure the cancellation of the IR enhancements, we
show that when the locality condition (2.39) is satisfied, the IRdiv, which comes from the
momentum integral, is cancelled in the correlation functions of the invariant variable under
the large gauge transformations. In the following, we denote a genuine gauge invariant
variable as gR(x) without specifying it. (One way to construct gR(x) was discussed in
HJEP10(27)
kL
kL
(4.1)
(4.2)
(4.3)
gR(x) = eiQζW (kL)gR(x)e−iQζW (kL) .
As in eq. (2.37), but focusing on a single soft mode, we expand the correlation function of
=
kL
0 = ζ˜c hΨ  iQζW (kL), gR(x1) · · · gR(xn)  Ψiζ˜c
kL
ζ˜c hΨ gR(x1) · · · gR(xn) Ψiζ˜c .
kL
hΨ gR(x1) · · · gR(xn) Ψi
kL
=
Z dζ˜kcLψ(ζ˜kcL)2 ζ˜c hΨ gR(x1) · · · gR(xn) Ψiζ˜c ,
where we noted that the soft mode ζ˜kL commutes with the genuine gauge invariant variable.
When the “locality” condition holds, since gR commutes with QζW (kL) we obtain
One important property of gR is being constructed only by local quantities such
that commute with the soft modes and remains invariant under the inhomogeneous
Since the correlator in eq. (4.3) is independent of the soft mode ζ˜kcL, it commutes with the
integral over ζ˜kcL. Then, the divergent integral R dζ˜kcLψ(ζ˜kcL)2 in eq. (4.2) simply becomes
the normalization factor, which should be canceled in computing connected diagrams. Here,
we picked up a certain wavenumber kL. However, repeating the same procedure for the
whole soft modes, we find that all the soft modes which correlate with the hard modes
are canceled out. This cancellation yields a suppression of the soft modes which interact
with the hard modes and ensures the absence of the IRdiv in the correlation function of
gR. In this way, we find that while the quantum state  Ψi is not invariant under the
inhomogeneous dilatation, which inserts the soft mode ζkL, the correlation function of gR
for the quantum state  Ψi is insensitive to the insertion. In ref. [41], this cancellation of
the correlation between the soft modes and the hard modes was presented by considering
the squeezed bispectrum.
Here, let us further discuss the relation between the genuine gauge invariance and the
absence of the IRdiv. Changing the boundary condition at the edge of the observable
region, we can alter the spatial average of the curvature perturbation ζ [39, 40, 60]. This
can be expressed as the dilatation whose constant parameter s is given by the spatial
average of ζ in the observable region. This dilatation changes the constant part of all
the modes with kL <∼ 1/LO, where LO is the size of the observable region, not only the
homogeneous mode with k = 0. Therefore, the genuine gauge invariance requires being
insensitive to the excitation of the constant soft modes ζkL. As we argued in section 2.3.2
(see around eq. (2.54)), quantum states which satisfy the locality condition do not preserve
the genuine gauge invariance in the sense that it is not insensitive to the insertion of the
constant part of the soft modes ζkL. What preserves the genuine gauge invariance is the
correlation function of the genuine gauge invariant operator evaluated for such quantum
states.11 While the relation between the dilatation invariance and the cancellation of the
IRdiv has been discussed in a number of literatures, e.g., in refs. [
39, 40, 44, 60, 63, 65–69
],
this aspect has not been clearly described elsewhere.
By contrast, since the curvature perturbation ζk is not a genuine gauge invariant
operator, the correlation functions which includes the operator ζk suffers from the IRdiv
(and also IRsec and SG). Because of that, the correlation functions hζkL OkS1 · · · OkSn i
diverge due to the accumulation of the soft modes. Here, OkS is either ζkS or OkS . In order
to make these correlation functions finite, we need to somehow introduce IR regularization.
However, recall that a naive introduction of the IR cutoff violates the dilatation invariance,
which was the starting point of the discussion about the consistency relation. Therefore,
HJEP10(27)
to be precise, we should not understand the consistency relation as the relation between
the correlation functions for the Heisenberg operators ζkL OkS1 · · · OkSn and OkS1 · · · OkSn ,
since they are not welldefined. Instead, the consistency relation we discussed in this paper
should be understood as the relation between the “correlation function” for the hard modes
OkS1 · · · OkSn without any propagation of the soft modes and the one with the additional
insertion of the free soft mode as an external leg. Then, both of them do not contain the
loop corrections of the soft modes, which can lead to the IR enhancements. This is the
reason why we needed to employ the approximation ≈ in deriving the consistency relation.
s.l.
5
Relevance and irrelevance of soft graviton insertion
The curvature perturbation ζ and the graviton γij are both massless fields and they have
similar IR behaviours. In this section, we briefly show that the discussion about the
consistency relation and the IR divergence for the graviton proceed almost in parallel to those
for the curvature perturbation ζ. For this purpose, we consider the shear transformation,
which is a large gauge transformation:
(5.1)
(5.2)
x
i
→ x˜i ≡ he S2 iij xj ,
where Sij is a constant symmetric and traceless tensor. Under this large gauge
transformation, the spatial metric transforms as
heγ˜(t, x˜)i
= he− S2 i k he− S2 i l heγ(t, x)i
ij
i
j
kl
.
At the linear perturbation, γij is shifted as γ˜ij = γij − Sij . The classical action is invariant
under the large gauge transformation (5.1).
11Although the locality condition is necessary condition for the absence of IRdiv, this does not
immediately imply that the locality condition is a requirement for the quantum state of the whole universe. When
we discuss observables for a local observer, it would be allowed to trace out the degrees of freedom which
the observer cannot see. After tracing out these degrees of freedom, the density matrix of the universe
will be block diagonalized with a good precision. Then, the observables will correspond to the expectation
values just for one of the blocks in the density matrix. In this sense, the actual observables are likely to be
quite different from the simple expectation values for a given wave functional of the whole universe.
Similar to the dilatation, we define the Noether charge for the large gauge
transformation (5.1) as
where πγij denotes the conjugate momentum of γij and
Qγ ≡ 2
d3x ΔSγij (t, x)πγij (t, x) + πγij (t, x)ΔSγij (t, x) ,
ΔSγij (t, x) ≡ γ˜ij (t, x) − γij (t, x) .
The invariance of the quantum state under this transformation requires
Repeating a similar argument, we find that the invariance is preserved, when the following
conditions
Qγ  Ψi = 0 .
∂
∂γ¯c ψ(γ¯c) = 0 ,
iQγ  Ψiγ¯c = Sij ∂γ¯icj  Ψiγ¯c ,
∂
jected quantum state into the eigenstate of γ¯icj .
are satisfied. Here, γ¯ij denotes the homogeneous mode of γij and  Ψiγ¯c denotes the
pro
Inserting the window function W (x) into the integrand of the Noether charge Qγ and
performing the Fourier transformation, we define QγW (kL), which inserts the soft graviton
γij,kL . Again, we find that when the condition (5.7) can be extended to the soft modes
with kL 6= 0, i.e.,
iQγW (kL) Ψiγ˜−ckL
= (2πLc)3Sij −kL ∂γ˜icj −kL
 Ψiγ˜−ckL
∂
with Sij kL ≡ Wˆ (kL)Sij , we obtain the consistency relation which relates the correlation
functions for the hard modes and those with single insertion of the Gaussian soft graviton
(see, e.g., refs. [14, 70]). The quantum state  Ψi changes due to the single insertion of the
soft graviton, i.e.,
QγW (kL) Ψi 6= 0 ,
because the wave function is not completely flat in the direction of the soft graviton γij kL
with kL 6= 0 in contrast to the homogeneous mode, whose wave function is completely
flat, satisfying eq. (5.6). Let us emphasize again that this is not a spontaneous symmetry
breaking, because the large gauge transformation (5.1) with an inhomogeneous Sij is not
a symmetry of the classical action.
The correlation function of the genuine gauge invariant operator gR is insensitive to
the insertion of the soft graviton. This ensures the absence of the IRdiv due to the soft
graviton in the correlation functions of gR evaluated for  Ψi. By contrast, when we evaluate
a correlation function for an operator which does not preserve the genuine gauge invariance,
the insertion of the soft graviton changes the correlation function and this can lead to a
break down of the perturbative expansion [71, 73]. (See also ref. [74].)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
HJEP10(27)
Concluding remarks
The relation among the large gauge transformations, the consistency relations for the soft
modes of ζ and γij, their conservation in time, and the IR enhancements has been discussed
in a number of literatures. However, as far as we understand, this relation was not fully
clarified and it was sometimes understood in a misleading manner. The purpose of this
paper is to sharpen the argument about the relation among these four. The invariance of
the quantum state  Ψi under the dilatation and shear transformations can be preserved,
when the following two conditions are fulfilled. First, the amplitude of the wave function
ψ should be flat towards the directions for the homogeneous modes of ζ and γij. Second,
operating the Noether charges Qζ and Qγ on the quantum state of hard modes is equivalent
to additively and timeindependently shifting the homogeneous modes ζ¯ and γ¯ij, which
interact with the hard modes, as described in eqs. (2.29) and (5.7).
The invariance under these large gauge transformations, just itself, leads to neither the
consistency relation nor the absence of IRdiv. The additional conditions (2.29) and (5.7) are
the nontrivial extensions to those which describe the insertion of the soft modes kL(6= 0),
which are not always satisfied. These conditions can be interpreted as the locality condition,
which states that the inhomogeneous dilatation and shear transformations only change
the values of ζ and γij within each local universe. Since the wave function is not flat
in the directions of the soft modes for the curvature perturbation and the graviton, the
quantum state  Ψi changes due to the insertion of these soft modes. When the locality
conditions (2.39) and (5.8) are satisfied, the influence of these soft modes are described
by the wellknown consistency relations under the Gaussian approximation of the wave
function for the soft modes. This argument also applies in deriving the consistency relation
for massive fields. We also showed that the locality condition (2.39) implies the conservation
of ζ in the soft limit within the perturbation theory. The same argument also applies to γij.
The final issue is the IR enhancements due to the soft modes of ζ and γij. In contrast
to the correlation functions for ζ and γij, which are not genuinely gauge invariant, the
correlation functions for a genuine gauge invariant operator remain invariant under the
insertion of the soft modes for the curvature perturbation ζ and the graviton γij, when
the locality conditions hold. This ensures the absence of the IRdiv due to the soft modes
of ζ and γij. In this paper, we did not discuss the IRsec and the SG, which yields the
secular growth. When the locality conditions hold at each time slicing, repeating the same
argument as in refs. [44, 64], we can show the absence of the IRsec and SG. In fact, this
is the case when we choose the adiabatic vacuum (or the Euclidean vacuum) as the initial
state of the universe.
Recently, the relation among the asymptotic symmetry, the soft theorem, and the IR
divergence was discussed about gauge theories in asymptotically flat spacetime [26–28, 32].
In ref. [26], it was shown that the Weinberg’s soft theorem for the soft photons and gravitons
can be derived as the WardTakahashi identities for the asymptotic symmetry. (For a recent
review, see ref. [29].) In ref. [32], the relation between the asymptotic symmetry and the
IR divergence of the QED was discussed. About the IR divergence in QED, Faddeev
and Kulish showed that the IR finiteness can be guaranteed, when we consider the dressed
charged particles by soft photon clouds. (See also refs. [76, 77].) While there is a qualitative
difference between the inout formalism in QED and the inin formalism in cosmology, our
genuine gauge invariant operator gR, whose correlators can be IR finite with an appropriate
choice of the initial state, also dresses the clouds of the soft ζ and γij as external legs.
At first glance, the IR structures for the gauge fields in the asymptotically flat
spacetime have a certain similarity to those for the primordial perturbations ζ and γij . However,
a closer look may also reveal some differences in these two cases. For the gauge theories
in the asymptotically flat spacetime, we are interested in the transition at the asymptotic
infinity between before and after the propagations of the soft photons and gravitons. On
the other hand, for the primordial perturbations, the asymptotic infinity is the spatial
infinity and is out of reach in a causal evolution. Instead, what we are interested in is a
locally defined quantity like an actually observable quantity. Moreover, as is summarized
above, in order to derive the consistency relation, the soft theorem for ζ and γij , we need
to assume the locality condition, which is not trivially satisfied even in a Diff invariant
theory, at least in the current gauge choice (the Maldacena gauge [14]). These give qualitative
differences from the gauge theories in the asymptotically flat spacetime, which trivially
satisfy the locality.
Acknowledgments
We would like to thank M. Mirbabayi for a fruitful conversation. T.T. was supported in part
by JSPS KAKENHI Grant Number JP 17H06357, 17H06358, 24103001 and 24103006, and
15H02087. Y.U. is supported by JSPS KAKENHI Grant Number JP 26887018, 16K17689,
and 16H01095. This research is supported in part by Building of Consortia for the
Development of Human Resources in Science and Technology and the National Science Foundation
under Grant No. NSF PHY1125915 and also by DAIKO FOUNDATION.
A
Perturbative and nonperturbative definition of Noether charge
In this section, we show that performing the dilatation and performing the perturbative
expansion are not commutable processes, i.e., the dilatation transformation in the
Heisenberg picture and the one in the interaction picture lead to different expressions. In order
to see this, let us first consider the dilatation in the Heisenberg picture with the use of the
Noether charge Qζ as
ζs(x) = eiQζ ζ(x)e−iQζ = ζ(x) + Δsζ(x) .
Then, we perturbatively expand both ζ(x) and ζs(x), i.e., before and after the dilatation
transformation, following the standard procedure, as
and
ζ(t, x) = U I†(t) ζI (t, x) U I (t) ,
ζs(t, x) = UsI†(t) ζsI (t, x) UsI (t) ,
(A.1)
(A.2)
(A.3)
Next, we show that the interaction picture field ζsI (t, x), which is related to ζs as in
eq. (A.3), does not coincide with
ζ˜sI (x) ≡ eiQIs ζI (x)e−iQIs = ζI (x) + ΔζI (x) ,
which is given by performing the dilatation transformation in the interaction picture. Here,
ΔζI is given by replacing the Heisenberg fields with the interaction picture fields in Δζ(x).
Notice that ζ˜sI (x) is related to ζs(x) by the unitary operator U I (t), i.e.,
ζ˜sI (x) = U I (t)ζs(t, x)U I†(t) ,
ζsI (t, x) = UsI (t)ζs(t, x)UsI†(t).
while the standard perturbative prescription in the frame after the dilatation
transformation uses the interaction picture field given by
As is shown in eq. (A.5), since the unitary operator UI (t) changes under the dilatation
transformation, we obtain
ζ˜sI (x) − ζsI (x) = O(s) .
Therefore, the dilatation transformation in the interaction picture (A.6) does not give the
interaction picture field defined in the standard prescription of perturbation theory after
the dilatation transformation, i.e., eq. (A.3). This discrepancy vanishes by sending s to 0.
where U I and UsI denote the unitary operators which relate the Heisenberg and interaction
picture fields for before and after the dilatation transformation. Using the interaction
Hamiltonian HI
≡ H − H0, the unitary operator is given by
U I (t) ≡ T e−i R t dt′HI ,
UsI (t) ≡ T e−i R t dt′HsI .
Since the free Hamiltonian H0 changes due to the dilatation transformation while the total
Hamiltonian does not, the interaction Hamiltonian HI and U I also change through the
UsI †(t)U I (t) = 1 + O(s) .
(A.5)
(A.4)
(A.6)
(A.7)
(A.8)
(A.9)
(B.1)
(B.2)
B
Computing the effective action
The linear term in the effective action is given by
In this appendix, we derive the expression of Wδ(gna)1 ···δgan (x1, · · · , xn), defined in eq. (3.17).
iSe′ff(1) = X Z
a=±
dd+1x δga(x)Wδ(g1α) (x) ,
where Wδ(g1α) is given by the expectation value as
Wδ(g1+) (x) = −Wδ(g1−) (x) =
* δiSχ
δg(x) δg=0
+
respect to δg+, we obtain
Wδ(g2+)δg˜+(x1, x2) = i2 * δSχ[δg+, χ+]
δg+(x1)
δSχ[δg+, χ+]
δg˜+(x2)
+ iδ(x1 − x2)
* δ2Sχ[ζ+, χ+]
δg+(x1)δg˜+(x1) δg+=0 ±
δg+=0 ±
+
,
where δg and δg˜ are either δN , Ni, ζ, or γij. Here, we introduced the expectation value:
R Dχ+ R Dχ− O[χ+, χ−]eiSχ[0, χ+]−iSχ[0, χ−]
the second variation of Se′ff with respect to δg− is given by
Since the action Sχ[δg+, χ+] includes only local terms, the variation of Sχ[δg+, χ+] with
respect to δg+(x1) and δg˜+(x2) yields the delta function δ(x1 − x2) in eq. (B.3). Similarly,
HJEP10(27)
Wδ(g2−)δg˜−(x1, x2) = i2 * δSχ[δg−, χ−]
δg−(x1)
δSχ[δg−, χ−]
* δ2Sχ[δg−, χ−]
δg−(x1)δg˜−(x1) δg−=0 ±
Taking the derivative with respect to both δg+ and δg−, we obtain
and
Wδ(g2+)δg˜−(x1, x2) = −i
Wδ(g2−)δg˜+(x1, x2) = −i
2 * δSχ[δg+, χ+]
δSχ[δg−, χ−]
δSχ[δg+, χ+]
When the interactions of χ are perturbatively suppressed, we can compute the functions
Wδ(g2a)1δg˜a2 (x1, x2) by expanding them in terms of the free propagators for χ.
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