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Published for SISSA by Springer
Received: March 31, 2021
Accepted: May 12, 2021
Published: May 27, 2021
Cosmological Cutting Rules
Scott Melville and Enrico Pajer
Department of Applied Mathematics and Theoretical Physics, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, U.K.
E-mail: scott.melville@damtp.cam.ac.uk, enrico.pajer@gmail.com
Abstract: Primordial perturbations in our universe are believed to have a quantum ori-
gin, and can be described by the wavefunction of the universe (or equivalently, cosmological
correlators). It follows that these observables must carry the imprint of the founding prin-
ciple of quantum mechanics: unitary time evolution. Indeed, it was recently discovered
that unitarity implies an infinite set of relations among tree-level wavefunction coefficients,
dubbed the Cosmological Optical Theorem. Here, we show that unitarity leads to a sys-
tematic set of “Cosmological Cutting Rules” which constrain wavefunction coefficients for
any number of fields and to any loop order. These rules fix the discontinuity of an n-loop
diagram in terms of lower-loop diagrams and the discontinuity of tree-level diagrams in
terms of tree-level diagrams with fewer external fields. Our results apply with remark-
able generality, namely for arbitrary interactions of fields of any mass and any spin with a
Bunch-Davies vacuum around a very general class of FLRW spacetimes. As an application,
we show how one-loop corrections in the Effective Field Theory of inflation are fixed by
tree-level calculations and discuss related perturbative unitarity bounds. These findings
greatly extend the potential of using unitarity to bootstrap cosmological observables and
to restrict the space of consistent effective field theories on curved spacetimes.
Keywords: Effective Field Theories, Classical Theories of Gravity
ArXiv ePrint: 2103.09832
Open Access, c© The Authors.
Article funded by SCOAP3. https://doi.org/10.1007/JHEP05(2021)249
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Contents
1 Introduction 1
1.1 Summary of results 3
2 Feynman rules for wavefunction coefficients 6
3 Some examples of cutting wavefunction diagrams 10
3.1 Cutting one propagator 10
3.2 Cutting two propagators 14
3.3 Cutting a loop 16
4 General cutting rules for a single scalar field 18
4.1 Lemma: a propagator identity 20
4.2 Proof of the cutting rules 24
4.3 Extension to multiple fields of any mass and spin 26
5 Inferring loops from trees using perturbative unitarity 27
5.1 On Minkowski 29
5.1.1 φ4 on Minkowski 29
5.1.2 φ3 on Minkowski 31
5.2 On de Sitter 33
5.2.1 φ˙3 on de Sitter 34
5.2.2 φ˙(∂iφ)2 on de Sitter 35
5.3 For the EFT of inflation 36
5.3.1 Wavefunction at one-loop 36
5.4 Physical interpretation 37
5.4.1 Power spectrum at one loop 39
6 Discussion 41
A Cutting rules from the Schrödinger picture 43
A.1 Tree-level constants of motion 45
A.2 Loop-level constants of motion 46
B List of propagator identities 48
B.1 Tree-level diagrams 48
B.2 Single-loop diagrams 49
C Explicit one-loop computation for p˙i3 49
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1 Introduction
Unitarity is a central pillar of quantum mechanics. On the one hand, the positive norm of
states in the Hilbert space is essential for ensuring that probabilities are positive. On the
other hand, a unitary time evolution ensures that the total probability is conserved and
hence the theory can make consistent statistical predictions for observables. In quantum
field theory on flat spacetime, several general properties and relations are known to follow
from unitarity (see e.g. [1]). For example, n-point correlators must factorize into products of
lower order ones in particular kinematic limits. Through the LSZ reduction formula, this in
turn leads to the factorization of amplitudes and the positivity of factorization coefficients.
An even more general consequence of unitarity is the Optical Theorem, which constrains
amplitudes for generic values of the kinematic variables. The non-linear nature of the
Optical Theorem is particularly useful in perturbation theory because it allows one to fix
higher-order amplitudes in terms of lower-order ones. In its most basic implementation, this
allows one to fix the imaginary part of one-loop diagrams in terms of tree-level ones. While
the Optical Theorem is a fully non-perturbative result, it is oftentimes useful to know how
it is satisfied order by order in perturbation theory — this is given by Cutkosky’s Cutting
Rules [2] (see also [3, 4] for a pedagogical derivation). In a nutshell, these rules tell us how
to compute the discontinuity of a given loop amplitude across one of its branch cuts using
some modified Feynman rules, in which the propagators of particles responsible for the
discontinuity are substituted with delta functions that put their four-momenta on-shell. It
is important to notice that in all of the above cases, one manages to express the rather
formal condition of unitary time evolution in terms of a constraint on physical observables,
namely amplitudes in this case.
Somewhat surprisingly, until a few months ago an analogous understanding of the im-
plications of unitarity was missing in the case of cosmological spacetimes and primordial
correlators. In this paper1 we fill this gap and derive Cosmological Cutting Rules, which,
in analogy with their flat space counterpart, consists of a set of unitarity conditions to be
satisfied order by order in perturbation theory. The natural observable for which these con-
ditions are formulated is the wavefunction of the universe. If desired these can be translated
into constraints on correlators. However, in the most general case (e.g. without restricting
to massless scalar field), the wavefunction expressions are much more compact. Our results
build upon a recently derived Cosmological Optical Theorem [6], and the associated con-
served quantities of [7]. The main insight of this work has been to recognize that the Her-
mitian conjugate time evolution, U †, appearing in the iconic unitarity condition UU † = 1,
can be related to a specific analytic continuation of the wavefunction of the universe, with
the same boundary conditions (the Bunch-Davies vacuum in most practical applications).
This suggests that unitarity fixes a very specific set of discontinuities, which is indeed what
we prove in the rest of the paper. This is highly non-trivial. Naively one might have ex-
pected that the imprint of quantum mechanics limits itself to the non-commutation of fields
with their conjugate momenta. If this were the case, unitarity would be a weak constraint
1A complementary discussion of cutting rules in cosmology will appear in [5] with emphasis on extensions
to massive and spinning fields beyond de Sitter at tree level, and a number of non-trivial checks.
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because the natural cosmological observables associated with the conjugate momenta decay
exponentially with (cosmological) time during inflation and are therefore practically unob-
servable. Instead, the Cosmological Optical Theorem tells us that the quantum mechanical
origin of perturbations manifests itself in a very specific analytic structure of the wavefunc-
tion. Recall that the boundary wavefunction encodes the correlation of fields at the same
time and at separated spatial points. From this point of view there isn’t a priori a natural
expectation of what unitarity would mean for such an object because time has completely
disappeared. This is in stark contrast with what happens in AdS, where the CFT on
the boundary still has a standard notion of time and of the associated unitarty evolution
(see [8, 9] for progress on the cutting rules in AdS). It is therefore quite remarkable to finally
discover how time evolution is hidden in the spatial correlation at the boundary of de Sitter.
One might hope that cutting rules in cosmology can be derived in complete analogy
with flat spacetime, but this is unfortunately not the case beyond tree-level diagrams. In
flat space, the cutting rules can be derived from a master “largest time equation” [3, 4, 10].
An analogous formula can be derived for the bulk-to-bulk propagator appearing in the
calculation of the wavefunction of the universe, a close relative of the Feynman propaga-
tor. However, such a procedure does not map directly to the standard representation of
wavefunction coefficient in terms of bulk time integrals. The main obstacle is that, when
computing a wavefunction either in Minkowski or in FLRW spacetimes, we need to adjust
the propagator to account for the presence of a boundary corresponding to the time at which
the wavefunction is computed. It is the presence of the associated boundary term in the
(bulk-to-bulk) propagator that makes the Cosmological Cutting Rule look quantitatively
different from their flat spacetime analogue. Away from the boundary, i.e. in the so-called
vanishing total energy limit, our cutting rules should reduce to the well-known ones for
amplitudes. However, the Cosmological Cutting Rules encode more information. Indeed,
as it will be discussed elsewhere, while one kinematical limit of the Cosmological Optical
Theorem produces the standard Optical Theorem, a different kinematical limit leads to the
factorization theorems at the heart of on-shell methods for amplitudes (see e.g. [11–13]).
It is interesting to ask which types of functions can appear in the final result for the
wavefunction coefficients in perturbation theory. For comparison, we know that amplitudes
at tree level only involve rational functions of the momenta (and the spinor helicity variables
for spinning fields). Logarithm, polyogarithm and their associated branch cuts appear at
loop level. Things are unfortunately more complicated in cosmology and the reason can be
traced back to the absence of time translation invariance (even the maximally symmetric
de Sitter does not have a globally defined time-like Killing vector). Indeed, even at tree
level, for a diagram with V vertices we have to perform V nested integrals in time and even
starting with simple mode functions such as those for massless and conformally couples
scalar fields (see (4.23)), we can end up with polylogarithms (see e.g. [14, 15]). From this
perspective, the Cosmological Cutting Rules can be thought of as identifying which parts of
wavefunction coefficients can be formulated in terms of “simpler” functions. In particular,
cutting rules tell us that a specific discontinuity can be computed in terms of diagrams
with one or more fewer time integrals, which feature functions with a lower transcendental
weight i.e. closer to the starting mode functions (e.g. in the sense of “the symbol” [15, 16]).
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Related to this, it would be interesting to see if our relations have a natural avatar in the
cosmological polytope representation of the wavefunction [16–19].
The Cosmological Cutting Rules we derive are a very useful practical tool to derive
certain effects of quantum loops while performing only tree level calculations. This is par-
ticularly useful in cosmology where, due to the absence of time translation invariance, cal-
culations become computationally demanding very quickly. For example, starting with [20]
much attention has been devoted to loop corrections during inflation. The simplest possible
case is a correction to the power spectrum, which at least naively has a chance to be sizable
in general class of models that are captured by the Effective Field Theory of inflation [21].
The cutting rules allow one to preform these calculation with much less effort than with
the direct bulk integration, as we will see in section 5.
Dulcis in fundo, we discuss the “bootstrap” approach, namely the prospect of using the
powerful constraints of unitarity, combined with other basic principles such as locality, the
choice of vacuum and symmetries as a computational tool to derive observables, and poten-
tially bypass the traditional bulk in-in calculation. This approach has a demonstrated track
record for the calculation of amplitudes [11–13], and has gained much traction in the cos-
mological context. Indeed, in the presence of a high degree of symmetry, such as Poincaré
invariance in Minkowski, very general results can be derived, such as for example the classi-
fication of all consistent cubic amplitude for particles of any spin (see e.g. [22–24]). Already
in this context, relaxing the amount of symmetry opens the door for many new and relative
unexplored possibilities. For example, in [25], all consistent cubic amplitudes were derived
allowing for spontaneously (non-linearly realized) or explicitly broken Lorentz boosts, as
relevant for many systems of interest including all conceivable cosmological backgrounds.
Similarly, when restricting to the most symmetric spacetime relevant for cosmology, namely
de Sitter, very general results can be obtained, as for example various combinations of scalar
and graviton correlators [26–36]. When combined with much progress on the front of per-
turbative calculations [14, 17, 37–43], these powerful symmetry-based results have given us
a much better understanding of general structures that appear in the wavefunction coeffi-
cients, such as singularities and the analytic structure. At the same, if we want to make
connection with observations we absolutely need a “boostless” bootstrap approach where
we relax the requirement of invariance under de Sitter boosts, since such symmetries are
incompatible with large non-Gaussianity in single field inflation [44]. Very promising re-
sults in this direction have already been derived using constraints from factorization [36],
the formulation of very general boostless Bootstrap Rules [45], and the recently derived
Manifestly Local Test and partial-energy recursion relations [46]. From this perspective
our Cosmological Cutting Rules add a powerful tool to bootstrap in full generality higher
order correlators form lower order ones, and in particular exchange and loop diagrams.
1.1 Summary of results
For the convenience of the reader, we provide below a summary of our main results.
• We derive Cosmological Cutting Rules for the wavefunction coefficients ψn for any
number of external legs and to any loop order. The rules are as follows (see section 4
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for a more formal discussion):
– For any particular diagramD that contributes ψ(D)n to a wavefunction coefficient,
sum over all possible 2I ways to cut its I internal lines. This may divide D into
a set of disconnected subdiagrams, each with associated ψ(subdiagram).
– Take the discontinuity (defined in (3.3)) of all possible subdiagrams by analyt-
ically continuing all external legs except those arising from the cutting of an
internal line.
– For every cut line add a factor of the power spectrum P , and then integrate over
all cut momenta (which now flow to the boundary).
Schematically, this procedure results in the following constraints, which we call Cos-
mological Cutting Rules (see (4.4))
i Disc
internal
lines
[
iψ(D)
]
=
∑
cuts
 ∏
cut
momenta
∫
P
 ∏
subdiagrams
(−i) Disc
internal &
cut lines
[
iψ(subdiagram)
]
, (1.1)
• Since sometimes a picture is worth a thousand words, we provide an example in
figure 1 below, where we state the Cosmological Cutting Rules graphically. The dou-
ble vertical red lines denote a cut. The discontinuity is taken of every disconnected
diagram with argument given by all highlighted lines (in orange). According to our
definition of Disc in (3.3), the arguments of Disc are just spectators, i.e. they are not
analytically continued. Throughout this work internal lines are never analytically
continued.
• We provide several explicit examples of the Cosmological Cutting Rules at tree level,
for one (see section 3.1 reproducing the Cosmological Optical Theorem of [6]) and
two (see section 3.2) internal lines, as well as two one-loop examples (see section 3.3).
• Our Cosmological Cutting Rules are valid very generally. In particular, they apply
to fields of any mass and spin with arbitrary interactions (provided they are local
in time and compatible with Hermitial analytiticy, which is the case for all common
interactions). To account for these cases, the above rules can be simply modified by
assuming that each internal or external momentum k or p carries additional quantum
numbers, such as the type of field, its spin and possible charges. By keeping all the
polarization tensors in the vertices, the derivation of the Cosmological Cutting Rules
reduces straightforwardly to the case of scalar fields. Indeed, our results are valid in a
large class of FLRW spacetimes, including de Sitter, slow-roll inflation and all power
law cosmologies, as long as all fields satisfy the Bunch Davies vacuum. These gener-
alizations are reviewed in section 4.3, and we refer the reader to [5] for more details.
• As applications of our newly derived relations, in section 5 we show how to compute
certain loop corrections from tree-level results in a series of physically relevant cases.
First, in Minkowski we compute the one-loop correction to the power spectrum from
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! ! !
! ! ! !
" ! !
! ! ! !
Figure 1. An example of the Cosmological Cutting Rules, applied to a particular diagram that
contributes to the wavefunction of the universe.
λφ4 and λφ3 interactions, respectively. Second, around quasi de Sitter spacetime, we
consider the leading cubic interactions in the effective field theory of inflation, and
compute the induced one-loop correction to the power spectrum. In the case of p˙i3,
we confirm the result of [47], while our results for p˙i∂pi2 are new.
• In the bulk of the paper we use the path integral representation of the wavefunction
of the universe, which allows us to find a general result to all loop orders. In ap-
pendix A, we present the connection with the Schrödinger picture of [7], and show
some examples of one-loop cutting rules.
Notation and conventions. The spatial Fourier transformation is defined by,
f(x) =
∫
d3k
(2pi)3 f(k) exp(ik · x) ≡
∫
k
f(k) exp(ik · x) , (1.2)
and commutes with time derivatives. Bold letters to refer to vectors, e.g. k, and we write
their magnitude as k ≡ |k|, which we refer to as an “energy”. A prime on a wave function
coefficient or correlator denotes that we have extracted the overall momentum-conserving
delta function,
ψn(k1, . . . ,kn) ≡ ψ′n(k1, . . . ,kn) (2pi)3δ3
(∑
ka
)
(1.3)
≡ ψ′n(k1, . . . ,kn) δ˜3
(∑
ka
)
. (1.4)
Note that, unfortunately, our conventions for ψn differ from those in [5, 6] by a minus sign,
ψheren = −ψtheren . We apologize for the inconvenience that this might cause.
When discussing functions of four momenta, such as ψ4(k1,k2,k3,k4), it will be con-
venient to use the variables,
ps = |k1 + k2| , pt = |k1 + k3| , pu = |k1 + k4| , (1.5)
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which are related by p2s + p2t + p2u =
∑4
a=1 k
2
a (so only six of the seven variables ka, ps, pt, pu
are independent). For general n-point wavefunction coefficients, we adopt a convention in
which ka label momenta of external legs, pa label the momenta of internal legs, and qa is
reserved for dummy integration variables (which arise after performing every cut). This
conventions is encoded in the way we write the arguments of the wavefunction coefficients,
namely
ψn = ψn( external energies ; internal energies ; contractions ) (1.6)
= ψn(k1, . . . , kn; p1, . . . , pI ;ka · kb,ka · (kb × kc),ka · (kb), . . . ) (1.7)
≡ ψn({k}; {p}; {k}) , (1.8)
where the last argument {k} denotes rotation-invariant contractions of the external mo-
menta with δij , ijk or with polarization tensors. We define the power spectrum as
Pqq′ := 〈φqφq′〉 = Pq δ˜3
(
q + q′
)
. (1.9)
2 Feynman rules for wavefunction coefficients
Consider a (d + 1)-dimensional conformally flat spacetime, ds2 = a2(η)(−dη2 + dx2). We
describe the state of the Universe and its fields, denoted collectively by φ, at conformal
time η using the wavefunction Ψη[φ] = 〈φ|Ψη〉, where |φ〉 is a field eigenstate. Starting
from an initial state |Ω〉 at early times, the state at a later time η0 is given by Ψη0 [φ] =
〈φ|U(η0,−∞)|Ω〉, where U(η2, η1) is the unitary operator that implements time translations
from η1 to η2. This can be computed using the path integral,
Ψη0 [φ] =
∫ Φ(η0)=φ
Ω at η→−∞
DΦ eiS[Φ] , (2.1)
where Φk(η) represents paths which coincide with |Ω〉 at early times and end on the con-
figuration φ at η0, and S[Φ] is the corresponding classical action.
Wavefunction coefficients. The wavefunction (2.1) (a functional of the fields φ) is
conveniently represented in terms of wavefunction coefficients,2
ψk1...kn(η0) =
1
Ψη0 [0]
δn
δφk1 . . . δφkn
log Ψη0 [φ]
∣∣∣∣
φ=0
, (2.2)
which are functions of time (and momenta) only. For brevity we will not explicitly write
the dependence on the time η0, at which the state is defined.
2Explicitly (the choice of sign is the same as in [48]),
Ψ[φ] ∝ exp
[
+
∞∑
n=2
∫
k1,...,kn
1
n!ψk1...knφk1 . . . φkn
]
.
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Propagators. The wavefunction coefficients (2.2) can be computed perturbatively in
a diagrammatic expansion analogous to the usual Feynman diagrams used to compute
the partition function (sometimes called Witten diagrams in analogy with the AdS/CFT
calculation [49]). To do this, one first identifies the classical field configurations (saddle
points of S[Φ]) which dominate the path integral. These solve the equations of motion
δS[Φ]/δΦk = 0 subject to the boundary conditions,
lim
η→−∞(1−i)
Φk(η) = 0 and Φk(η0) = φk , (2.3)
which corresponds to projecting onto the free vacuum in the asymptotic past.3 Writing
the variation δS[Φ]/δΦk as,
Ok(η)Φk(η) = −δSint[Φ]
δΦk(η)
, (2.5)
where Ok(η)Φk denotes the linearized (exactly solvable) equations of motion and depends
only on the magnitude of the momentum, solutions can be constructed perturbatively in
the interactions Sint. This requires two propagators: the bulk-to-boundary propagator Kk
and the bulk-to-bulk propagator Gk, which satisfy,
Ok(η)Kk(η, η0) = 0 , Ok(η)Gk(η, η′, η0) = −δ(η − η′) , (2.6)
subject to the boundary conditions,
lim
η→η0
Kk(η, η0) = 1, lim
η→−∞(1−i)
Kk(η, η0) = 0 , (2.7)
lim
η→η0
Gk(η, η′, η0) = 0, lim
η→−∞(1−i)
Gk(η, η′, η0) = 0 ,
and the symmetry condition Gk(η, η′, η0) = Gk(η′, η, η0). The classical field configurations
are then defined implicitly by the relation,
Φk(η) = Kk(η, η0)φk +
∫
dη′Gk(η, η′, η0)
δSint[Φ]
δΦk(η′)
, (2.8)
which can be solved perturbatively to any desired order in the interactions. As an aside,
notice that Kk(η) is completely analogous to the transfer functions and growth functions
used in the study of perturbations of the large scale structures or of the cosmic microwave
background. It would be interesting to see if the techniques developed here could be also
useful in those lines of research.
3The free vacuum is annihilated by aˆk ∝ ωkφˆ−k + iΠˆ−k, and so this condition can also be written as,
lim
η→−∞
(Π−k − iωkΦ−k) = 0 , (2.4)
where Πk is the conjugate momentum associated with the path Φk. For a canonical scalar field of mass m,
Πk = ad−1∂ηΦk and ω2k = ad+1(k2/a2 + m2) in (2.4), which selects the behaviour Φk(η) ∼ exp(+ikη) at
early times.
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Connection with Feynman diagrams. The wavefunction coefficients (2.2) can then
be represented as a sum over diagrams, in which vertices correspond to the interactions in
Sint, and lines correspond to factors of either Kk(η, η0) (if connected to the final time η0) or
Gk(η, η′, η0) (if connected between two earlier times, η and η′ both < η0). These diagrams
are a close analogue of the Feynman diagrams which are used to represent time-ordered
correlation functions,
〈Ω|T φ(x1) . . .φ(xn)|Ω〉
〈Ω|Ω〉 =
(−i)n
Z[0]
δnZ[J ]
δJ(x1) . . . δJ(xn)
∣∣
φ=0 where Z[J ] =
∫
DφeiS[φ]+i
∫
x
J(x)φ(x) .
(2.9)
While these matrix elements are obtained by summing over all Feynman diagrams, replac-
ing Z[J ] withW [J ] = logZ[J ] generates instead the connected correlation functions, which
correspond to summing over only connected Feynman diagrams. In these diagrams, vertices
are the interactions contained in Sint, and the edges are either external lines (connected to
one of the xn), or internal lines, which correspond to the matrix elements,
External: 〈Ω|aˆkφˆk(η)|Ω〉′ = f∗k (η) ,
Internal: 〈Ω|T φˆk(η)φˆ−k(η′)|Ω〉′ = ∆k(η, η′) , (2.10)
where f∗k (η) and ∆k(η, η′) are the usual mode function and Feynman propagator respec-
tively. These Feynman rules reproduce the more laborious calculation of canonical quanti-
sation using φˆk(η) = fk(η)aˆ−k + f∗k (η)aˆ
†
k in the Heisenberg picture.
Comparing (2.9) with (2.2), we see that ψk1...kn corresponds to the connected part of
the matrix element,
〈0η0 |Πˆk1 . . . Πˆkn |Ψη0〉
〈0η0 |Ψη0〉
= (−i)
n
Ψη0 [0]
δnΨη0 [φ]
δφk1 . . . δφkn
∣∣∣∣
φ=0
, (2.11)
where |0η0〉 is the field eigenstate in which all fields are set to zero at time η0. Just as
the time-ordered correlators (2.9) can be represented via Feynman diagrams, so too can
the wavefunction coefficients. The only difference is that the rules for replacing inter-
nal/external lines (2.10) must be updated to,
Bulk-to-boundary: 〈0η0 |Πˆk(η0)φˆk(η)|Ω〉 = Kk(η, η0) =
f∗k (η)
f∗k (η0)
,
Bulk-to-bulk: 〈0η0 |T φˆk(η)φˆ−k(η′)|Ω〉′ = Gk(η, η′, η0) . (2.12)
Note that Gk(η, η′, η0) is similar to the Feynman propagator ∆k(η, η′), only with the feature
that it vanishes if either η or η′ are taken close to η0 (due to the zero-field eigenstate bra).
From (2.10) and (2.12), we see that the internal (bulk-to-bulk) propagators can be
written in terms of the external (bulk-to-boundary) propagators,4
∆k(η1, η2) = θ(η1 − η2)fk(η1)f∗k (η2) + (η1 ↔ η2) ,
Gk(η1, η2, η0) = 2Pk [θ(η1 − η2)Kk(η2)ImKk(η1) + (η1 ↔ η2)] (2.13)
= iPk [θ(η1 − η2)K∗k(η1)Kk(η2) + θ(η2 − η1)K∗(η2)K(η1)−K(η1)K(η2)] ,
4This expression for Gk is valid only for real momenta, k ∈ R. This is sufficient for this paper where we
never analytically continue internal energies.
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where Pk = 〈φˆkφˆ−k〉′ = |fk(η0)|2 is the power spectrum at the time η0 at which the state
is defined. Just as with ψk1...kn , above and in the following we do not explicitly write the
dependence on η0 in Kk, Gk or Pk.
In particular, the bulk-to-bulk propagator differs from the Feynman propagator by a
boundary term,
Gk(η1, η2) = i∆k(η1, η2)− iPkKk(η1)Kk(η2) . (2.14)
The presence of this additional term has a profound meaning and important practical
consequences. Its meaning is that we are in the presence of a (conformal) boundary.
We would find such a term in Minkowksi as well if we wanted to compute wavefunction
coefficients (or correlators) on a constant time hypersurface. It reminds us of the asymmetry
between the past and the future, which qualitatively distinguishes cosmology from particle
physics. This boundary term is the main obstacle to extend flat space cutting rules to
cosmology. While Veltman’s largest time equation still holds, it does not map explicitly to
a set of relation among observables. In this paper we overcome this difficulty by deriving
a bespoke set of Cosmological Cutting Rules.
Rules for computing a diagram. In the following we will express various contributions
to the wavefunction coefficients diagrammatically, in a way that is analogous to the usual
Feynman diagram expansion for amplitudes (see figure 2). The analogue of Feynman rules
are the following:
• Draw a graph and assign momenta ka to each of its n external legs and momenta pm
to each of the |I| internal legs in way that respects momentum conservation at every
vertex (but not energy conservation). For a diagram with L loops, this fixes all but
L internal “loop” momenta. Assign a conformal time ηA to each of the V vertices.
• Multiply a bulk-to-boundary propagatorKka(ηA) for every external leg (which reaches
the (conformal) boundary η0 → 0), and a bulk-to-bulk propagator Gpm(ηA, ηB) for
every internal line (that connects two vertices at times ηA and ηB).
• For every vertex at ηA, add the appropriate factors of momenta corresponding to
spatial derivatives and for time derivatives act with ∂ηA on the appropriate internal
or external line connected to the vertex. Sum over all allowed permutations. There
is no factor of i associated with the vertex. For example, the vertex corresponding
to λφn/n! is simply λ.
• Multiply by an overall factor of i1−L. This could equivalently be viewed as iV−I , a
factor of i for each vertex and a (−i) for each propagator, which accounts for the fact
that our normalisation of Gk in (2.6) differs from the usual Feynman normalisation.
• Integrate over all times ηA from −∞(1 − i) to η0 → 0 and over all loop momenta
pl ∈ R3.
Our strategy will be to first prove the cutting rules for individual diagrams, since
they can then be applied to any ψk1...kn to any desired order in perturbation theory (see
appendix A for an alternative derivation of the cutting rules directly at the level of the
ψk1...kn).
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!!!"" " ##$"% ""& !
""" '#$"% "$% ""&"$! ! !%"" " (&)*+,(+'! " (+'" " -(+'# "!- !&
Figure 2. A graphical representation of the Feynman rules to compute the wavefunction of the
universe in perturbation theory.
3 Some examples of cutting wavefunction diagrams
Our goal in this section is to present the algebraic structure of simple diagrams and how
one can compute certain discontinuities using the Cosmological Cutting Rules. The idea
is to see the practical application of these rules in concrete cases before moving on to the
move formal proof to all orders in section 4. We will start with the simplest case of a single
propagator and re-derive the Cosmological Optical Theorem of [6]. Then we will consider
how to cut two propagators, in turn at tree level and at one loop level. A parallel derivation
using the Schrödinger equation along the lines of [7] is presented in appendix A.
3.1 Cutting one propagator
For our first example, consider a simple cubic interaction, Sint[φ] =
∫
dtd3x a3(t)λφ3. The
corresponding wavefunction coefficients (with overall momentum-conserving delta functions
removed) are given by,
!! !"!# !$ "%#&!&"&#&$'%(
!# !$
#&!&"&#
!!
ψ′k1k2k3 = iλ
∫ t0
−∞
dtKk1(t)Kk2(t)Kk3(t)
ψ
(s) ′
k1k2k3k4 = iλ
2
∫ t0
−∞
dtL
∫ t0
−∞
dtRKk1(tL)Kk2(tL) Gps(tL, tR)Kk3(tR)Kk4(tR)
(3.1)
at tree level, as shown in the diagram above (ψ4 = ψ(s)4 + ψ
(t)
4 + ψ
(u)
4 , but we will focus on
the s-channel diagram). Note that the two time integrals in ψ(s)4 are nested: they do not
factorise since the bulk-to-bulk propagator Gps(tL, tR) contains a step function θ(tL− tR).
However, from (2.13) we see that the imaginary part,
ImGps(tL, tR) = 2Pps ImKps(tL) ImKps(tR) (3.2)
factorises into separate functions of tL and tR. Consequently, if we can extract the imagi-
nary part of the internal line in the ψ(s)4 exchange diagram, then the two time integrals will
factor into a simple product ψ3×ψ3. This is achieved by evaluating ψn at a modified value
k¯ of the external energies, defined such that Kk¯(t) = K∗k(t), and applying a parity transfor-
mation on all internal and external spatial momenta, {k,p} → {−k,−p}. For example, for
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de Sitter mode functions for a massless or conformally coupled field with a Bunch Davies
vacuum, one has simply k¯ = −k, with the negative real k-axis being approached from
the lower-half complex plane to guarantee appropriate convergences [6]. Furthermore, to
simplify our notation, we’ll make often use of the following “discontinuity” operation,5
Disc
k1...kj
f(k1, . . . , kn; {p}; {k}) (3.3)
≡ f(k1, . . . , kn; {p}; {k})− f∗(k1, . . . , kj ,−kj+1, . . . ,−kn; {p};−{k}) ,
where {p} denotes internal energies, which are untouched by the Disc, and {k} are all
spatial momenta. In words, the Disc operation corresponds to subtracting the complex
conjugate of ψn with all external energies analytically continued to minus themselves except
for those listed in the subscript of Disc and all spatial momenta (internal or external)
reversed by parity k→ −k. For example, no subscript corresponds to replacing all ka → k¯a.
This Disc operation can be used to pick out the imaginary part of the corresponding internal
propagator,6
iDisc
ps
[
iψ
(s)
k1k2k3k4
]
= 2λ2
∫ t0
−∞
dtL
∫ t0
−∞
dtRKk1(tL)Kk2(tL) ImGps(tL, tR)Kk3(tR)Kk4(tR)
=
∫
qq′
iDisc
q
[
iψk1k2q
]
Pqq′ iDisc
q′
[
iψq′k3k4
]
, (3.4)
where we have used (3.2), and introduced the power spectrum Pqq′ which includes the
momentum conserving δ-function as in (1.9). Note that for translationally invariant inter-
actions each ψn also contains an overall momentum conserving δ-function, which in this
case can be used to set q′ = −q = ps.
We depict the cutting rule (3.4) diagrammatically as follows:
!! !"!# !$ !
"%%!
#&
!! !"!# !$ $ $'
% &'()(" %*)#)$)%)&*&+
% &'()% %*)#)$% "%%! % &'()%! %*%!)%)&
"
On the left-hand side, we have introduced a highlighted line to represent the propagator
of which we are extracting the imaginary part. The energy of the highlighted line appears
in the argument of Disc and so it not analytically continued. Our cutting rule then relates
5We use this terminology by analogy with the amplitude discontinuity DiscsA12→34 = 12i
(
A12→34 −
A∗34→12
)
, which appears in the flat space optical theorem.
6Note that (−i)Disc [iψn] = iDisc [cn] in the notation of [7].
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this to the first diagram shown on the right, where the red vertical lines indicate which
propagators are to be “cut”. By definition, when a propagator is “cut” we replace it with
two bulk-to-boundary propagators and insert a factor of the power spectrum, which is
shown in the final diagram on the right-hand side. This reproduces the Cosmological
Optical Theorem of [6] from the point of view of cutting rules.
An analogous relations holds for the t- and u-channels, and so the full wavefunction
coefficient obeys,
i Disc
psptpu
[iψk1k2k3k4 ] =
3∑
perm.
∫
qq′
iDisc
q
[
iψk1k2q
]
Pqq′ iDisc
q′
[
iψq′k3k4
]
(3.5)
at tree level (note that ψ(s)4 depends on pt and pu only through analytic combinations like
pt · pt and pt · pu, so Discpsptpu [iψ(s)4 ] = Discps [iψ(s)4 ], and similarly for ψ(t)4 and ψ(u)4 ).
Note that the role of the Disc combination is to take the imaginary part of the inter-
nal lines (bulk-to-bulk propagators) without affecting any external line (bulk-to-boundary
propagator). The external lines therefore only appear in our cutting equations as an overall
factor. For instance, (3.2) is also the relevant cutting rule for any diagram in which a single
internal line is connected to nL external legs on the left and nR external lines on the right,
providing we replaceKk1(tL)Kk2(tL) withKk1(tL) . . .KknL (tL) and replaceKk3(tR)Kk4(tR)
with Kk′1(tR) . . .Kk′nR (tR). The cutting rule for this diagram is the straightforward exten-
sion of (3.4),
iDisc
p
[
iψ
(p)
{k}{k′}
]
=
∫
qq′
iDisc
q
[
iψ{k}q
]
Pqq′ iDisc
q′
[
iψq′{k′}
]
, (3.6)
where p = ∑nLa ka = −∑nRa k′a is the total momentum flowing from the boundary into
(out of) the interaction vertices, i.e. the momentum carried by the internal line, and the ψ’s
on the right-hand side are contact diagrams with (nL + 1) and (nR + 1) legs, respectively.
We can therefore focus on only the internal lines (suppressing any external line factors),
since this provides more compact expressions which are applicable to a wider range of di-
agrams (i.e. any diagram in which an arbitrary number of external lines is attached to any
of the vertices).
Single-cut rules. Finally, note that although we focused above on a simple diagram with
only a single internal line, more generally in a diagram with many internal lines we can
always use an appropriate Disc to cut any single propagator. For instance, for the cubic
interaction considered in (3.1), one diagram which contributes to the quintic wavefunction
coefficient is given by
!!
""
!! !" !#
"# !$
"% "& "'
ψ
(pL, pR) ′
k1k2k3k4k5 = iλ
2
∫ t0
−∞
dtL dtM dtRKk1(tL)Kk2(tL)GpL(tL, tM )
×Kk3(tM )GpR(tM , tR)Kk4(tR)Kk5(tR) ,
(3.7)
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where pL = |k1 + k2|, and pR = |k4 + k5| are the momenta flowing through two internal
lines, which connect interaction vertices at times tL, tM and tR. To cut the GpL(tL, tM )
internal line, we take Disc
pL
[
iψ
(pL, pR)
k1k2k3k4k5
]
, which extracts the ImGpL(tL, tM ) and allows us
to use the propagator identity (3.2). Diagrammatically, this corresponds to,
!!
""
!! !" !#
"# !$
!
" #%!%!"$! $!&
"' "( ")
"""' "# "( ")
!$
which represents the single-cut rule discussed in [5],
iDisc
pL
[
iψ
(pL, pR)
k1k2k3k4k5
]
=
∫
qLq′L
iDisc
qL
[
iψk1k2qL
]
PqLq′LiDiscq′L
[
iψ
(pR)
q′Lk3k4k5
]
(3.8)
where ψ(pR)q′Lk3k4k5 is the exchange diagram with pR = k4 + k5 flowing through the in-
ternal line. Note that there are three δ-functions on the right-hand-side, which enforce
momentum-conservation at each vertex, namely qL = −k1 − k2 and q′L = −k3 − k4 − k5,
as well as overall momentum conservation k1 + k2 + k3 + k4 + k5 = 0. We will not discuss
these single-cut rules any further here, but refer the reader to [5] for a detailed analysis.
Before proceeding, it is worth commenting on the difference between the above single-cut
rule and the (multi-cut) cutting rules we will discuss in the rest of this paper:
• In single-cut rules we have to analytically continue all internal lines that are not cut,
in addition to the external lines. To make this possible, one needs to choose variables
such that the energies of all non-cut internal lines appear in the argument of ψ, so
that they can be analytically continued by Disc. Since the energies flowing in the
internal lines depend on the specific diagram chosen (i.e. the different channels), it
follows that the choice of variables for single-cut diagrams are diagram dependent.
This is in contrast with the cutting rules we discuss in this paper, in which case we
never analytically continue any internal line, and so it does not matter if its energy
appears or not as a variable. Indeed, notice that in all our examples, the internal
lines are either highlighted, therefore they appear in the argument of Disc and are
not analytically continued, or they are cut.
• In single-cut rules we can cherry-pick where to cut a given diagram. Conversely, for
the cutting rules in this work one has always to sum over all possible cuts, including
multiple cuts and no cuts at all. We will see this in the next subsection.
• Importantly, in their current formulation, single-cut rules apply only to tree-level
diagrams. The reason is that in a loop diagram, the momentum of some internal
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line is integrated over and so it is not clear how one could analytically continue it
by altering the variables of ψ. Conversely, the cutting rules we discuss here apply
to diagrams of any loop order. This is possible because, as we stressed above, we
never analytically continue any internal energy. We will see how to deal with loops
in section 3.3.
3.2 Cutting two propagators
Taking a closer look at the quintic wavefunction coefficient (3.7), we see that it contains
an integral of the form,∫ t0
−∞
dtL
∫ t0
−∞
dtM
∫ t0
−∞
dtRGpL(tL, tM )GpR(tM , tR)
[ 5∏
a=1
Kka
]
, (3.9)
which does not factorise due to the pair of θ(tL− tM ) and θ(tM − tR) functions within the
bulk-to-bulk propagators. As shown above, by taking a suitable imaginary part of (3.9)
(i.e. the Disc of the corresponding wavefunction coefficient), we can remove at least one of
these θ-functions, leading to the single-cut rule in (3.8), which now contains only a single
exchange integral.
Remarkably, there is another way to remove at least one θ-function from (3.9), and
that is to take the imaginary part of both propagators:
Im [GpL(tL, tM )GpR(tM , tR)] =
+ 2PpL Im [KpL(tL)] Im [KpL(tM )GpR(tM , tR)]
+ 2PpR Im [KpR(tR)] Im [KpR(tM )GpL(tL, tM )]
− 4PpLPpR Im [KpL(tL)] Im [KpL(tM )KpR(tM )] Im [KpR(tR)] . (3.10)
This factorises the three nested integrals (3.9) into the product of two or three lower n-point
coefficients. Using the Disc to pick out the imaginary part of the two internal propagators,
we can use (3.10) to factorise the 5-point coefficient (3.7) into products of lower n-point
functions, in particular ψ4 × ψ3 and ψ3 × ψ3 × ψ3,
iDisc
pLpR
[
iψ
(pL, pR)
k1k2k3k4k5
]
=
∫
qLq′L
iDisc
qL
[
iψk1k2qL
]
PqLq′LiDiscq′LpR
[
iψ
(pR)
q′Lk3k4k5
]
+
∫
qRq′R
iDisc
pLqR
[
iψ
(pL)
k1k2k3qR
]
PqRq′RiDiscq′R
[
iψq′Rk4k5
]
−
∫
qLq′L
qRq′R
iDisc
qL
[
iψk1k2qL
]
PqLq′LiDiscq′LqR
[
iψq′Lk3qR
]
PqRq′RiDiscq′R
[
iψq′Rk4k5
]
(3.11)
where ψ(pL)k1k2k3qR is the particular exchange contribution to ψ4 in which the internal line
carries momentum pL = k1 +k2. The cutting rule (3.11) corresponds to summing over all
possible cuts of the internal lines (the left-hand side corresponding to zero cuts), where a
cut bulk-to-bulk propagator is replaced by two bulk-to-boundary propagators and a factor
of the boundary power spectrum.
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Diagrammatically, we represent the cutting rule (3.11) as:
!!
""
!! !" !#
"# !$
! ! !
" ! !#%!%!"$! $!& $$ $$&#%#%#" #%!%!" #%#%#"
"' "( ")
$$ $$&$! $!&"""' "# "( ") "" "' "# ")"(
"#"""' ")"(!$ !!
These three cut diagrams on the right-hand-side correspond to taking discontinuities of each
disconnected subdiagram (which now contain at most a single bulk-to-bulk propagator),
and correspond to the three terms on the right-hand-side of the cutting rule (3.11). Note
that when we highlight two or more lines in any disconnected subgraph, it corresponds to
taking a single Disc in such a way that a single imaginary part is taken of the product of the
highlighted propagators (and should not be confused with taking multiple discontinuities
to extract multiple imaginary parts). For instance, the final diagram on the right-hand-side
has three disconnected components (so is the product of three separate Disc’s), and the
central subdiagram is given by Disc
q′LqR
[
iψq′Lk3qR
]
, which extracts the imaginary part of the
product Kq′LKqR .
Just as when cutting a single propagator, here as well it is only the external lines that
are analytically continued and not the internal lines. The cutting rule (3.11) can therefore
be easily generalised to any diagram which contains two internal lines connected in this
way. For instance, consider the diagram with three interactions vertices, VL, VM and VR,
shown below. A collection of external lines (with momenta {kL}) are connected to the left
interaction vertex VL, and similarly for vertices VM and VR. Internal bulk-to-bulk lines
connect VL to VM and VM to VR, and carry momenta pL and pR respectively. We denote
this particular diagram by ψ(pL, pR){kL}{kM}{kR}, and it contains a triple (nested) time integral
of the form (3.9). In general this integral can be difficult to perform exactly, however the
relation (3.10) allows us to express its discontinuity at fixed pL and pR in terms of objects
that involve only double time integrals. Explicitly, this gives the cutting rule,
iDisc
pLpR
[
iψ
(pL pR)
{kL}{kM}{kR}
]
=
∫
qLq′L
iDisc
qL
[
iψ{kL}qL
]
PqLq′LiDiscq′LpR
[
iψ
(pR)
q′L{kM}{kR}
]
+
∫
qRq′R
iDisc
pLqR
[
iψ
(pL)
{kL}{kM}qR
]
PqRq′RiDiscq′R
[
iψq′R{kR}
]
−
∫
qLq′L
qRq′R
iDisc
qL
[
iψ{kL}qL
]
PqLq′LiDiscq′LqR
[
iψq′L{kM}qR
]
PqRq′RiDiscq′R
[
iψq′R{kR}
]
. (3.12)
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3.3 Cutting a loop
The cutting rules (3.4), (3.5), (3.6), (3.8), (3.11) and (3.12) shown above are relations among
exclusively tree-level wavefunction coefficients. We will show how the Disc operation (3.3)
can also be used to reduce simple one-loop diagrams to a product of tree-level diagrams.
One-propagator loop. The simplest one-loop diagram contains a single internal line, as
shown below. Unlike the tree-level examples above, this single propagator is evaluated at
coincident times, Gp(t, t). In this case, it is not only the imaginary part of the propagator
which factorises, but also the real part,7
Re [Gp(t, t)] = Pp Im [Kp(t)Kp(t)] . (3.13)
This means that, considering the 1-loop contribution to ψn from the interaction λφn+2,
ψ1-loop ′k1...kn = λ
∫ t0
−∞
dt Kk1(t) . . .Kkn(t)
∫
p
Gp(t, t) (3.14)
we can use (3.13) to write its discontinuity in terms of a tree-level coefficient,
iDisc
[
iψ1-loopk1...kn
]
=
∫
qq′
(−i)Disc
qq′
[
iψtreek1...knqq′
]
Pqq′ (3.15)
where,
ψtree ′k1...knqq′ = iλ
∫ t0
−∞
dt Kk1(t) . . .Kkn(t)Kq(t)Kq′(t) , (3.16)
is the contact contribution to ψn+2. Diagrammatically,
! "#$% !& !"#$%%&
'! "#$% !& ! ''!()** (''!
!
+
!"#$
) ),
(''!
"#$"
% %&
7Note that since we have written the time-ordering in (2.10) and (2.12) as, T φˆk1(t1)φˆk2(t2) =
θ(t1 − t2)φˆk1(t1)φˆk2(t2) + θ(t2 − t1)φˆk2(t2)φˆk1(t1), we are treating θ(0) = 1/2.
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Some comments about the cutting rule (3.15):
(i) As in the previous examples, the cutting rule has effectively removed the need to per-
form an additional time integral — if ψtreen+2 has already been computed using (3.16),
then (3.15) can be used to infer Disc
[
iψ1-loopn
]
without ever carrying out the time
integral in (3.14).
(ii) Unlike in the tree-level examples, there are no longer enough δ-functions to fix all
of the internal momenta in (3.15), so one momentum integral is left over. However,
unlike in (3.14), the remaining momentum integral in (3.15) is now finite, and does
not require any regularisation.
(iii) One could also have used (3.2) to write this in terms of the tree-level result, but
since (3.2) involves two imaginary parts this would result in a cutting rule with
multiple (overlapping) discontinuities — these will be discussed separately elsewhere.
Two-propagator loop. The next-simplest loop diagram contains a single loop composed
of two internal lines, as shown below. In this case, the diagram contains two time integrals
over the product Gq1(t1, t2)Gq2(t1, t2),
ψ
1-loop, (p) ′
{k}{k′} = λ
2
∫ t0
−∞
dt1
∫ t0
−∞
dt2
∏
kj
Kkj (t1)
∏
k′j
Kk′j (t2) (3.17)
×
∫
p1p2
Gp1(t1, t2)Gp2(t1, t2) δ˜3(p1 + p2 − p) .
where p = ∑j kj is the momentum flowing into the loop.8 To factorise this into two
separate integrals, one can use the following identity,
2Re [Gp1(t1, t2)Gp2(t2, t1)] = 2Pp2 Im [Kp2(t1)Gp1(t1, t2)Kp2(t2)]
+ 2Pp1 Im [Kp1(t2)Gp2(t2, t1)Kp1(t1)]
− 4Pp1Pp2Im [Kp1(t1)Kp2(t1)] Im [Kp2(t2)Kp1(t2)] (3.18)
which relates the real part of the propagators to products of Kk and Gk, and which crucially
contains imaginary parts acting only on factors evaluated at the same times. This allows
one to write each of the terms on the right-hand side of (3.18) in terms of a single Disc
acting on a tree-level wavefunction coefficient,
iDisc
p
[
iψ
1-loop, (p)
{k}{k′}
]
=
∫
q2q′2
Pq2q′2(−i) Discp1q2q′2
[
iψ
tree, (p1)
{k}q2q′2{k′}
]
+
∫
q1q′1
Pq1q′1(−i) Discp2q1q′1
[
iψ
tree, (p2)
{k}q1q′1{k′}
]
(3.19)
+
∫
q1q′1
q2q′2
iDisc
q1q2
[
iψtree, contact{k}q1q2
]
Pq1q′1Pq2q′2 iDiscq′1q′2
[
iψtree, contactq′1q′2{k′}
]
8Note that each of these internal lines may correspond to different fields, but this can be viewed as simply
adding additional quantum numbers to the labels p1 and p2, i.e. in (3.18), the Ppj , Kpj and Gpj factors
correspond to either exchanged field 1 or field 2, and which one can be inferred from their momentum label.
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where p = ∑j kj is the momentum flowing into the loop from the boundary, and ψtree, (p2){k}qq′{k′}
corresponds to the diagram in which a propagator Gp2(t1, t2) connects external legs with
momenta {k} and q at time t1 to external legs with momenta {k′} and q′ at time t2.
Diagrammatically,
!! !! !"!! !"
!! "#" #!!! "#
$"
$""""#
%# %$%! !&
!! "#" #!!! "#
$"
$"$"$#
%& %'%! !#
!! "#" #!!! "#
$"
$""""#%&! %#!! "#" #!!! "#
$"%&%%#%
$"$"$#
" "
"# "
Note that at tree-level we had a choice about how many internal lines to highlight
with the Disc, leading to the single-cut rules (3.8) of [5] or our (multiple-cut) cutting
rules (3.11). At loop-level, it is no longer possible to use the Disc operation to extract
arbitrary imaginary parts — for the one-loop example above, the whole loop must be
highlighted, since it is not possible to extract ImGp1(t1, t2) alone. This is why going
beyond tree-level requires going beyond the cutting of single lines. In the following, we will
focus only on diagrams in which every internal line is highlighted, i.e. we never analytically
continue internal lines
To sum up, we have used simple algebraic relations between the bulk-to-bulk and
bulk-to-boundary propagators to derive powerful cutting rules which relate higher n-point
wavefunction coefficients to lower n-point coefficients, and which crucially can relate 1-loop
diagrams to (products of) tree-level diagrams. These relations turn out to be surprisingly
universal, and we will now show that, faced with any L-loop diagram, one can take appro-
priate discontinuities to reduce it to combinations of lower-point (L− 1)-loop diagrams.
4 General cutting rules for a single scalar field
In this section, we begin by stating and proving the cutting rules for a general diagram,
with any number of internal/external lines and with any number of loops, but focusing
on a single scalar field. We will prove this result with the help of an algebraic identity
for the imaginary part of the product of bulk-to-bulk propagators. In the next section we
generalize our result to multiple fields with any spin (section 4.3).
To help intuition, let’s begin with the following simplified statement of the cutting
rules:
i Disc
internal
lines
[
i ψ(D)
]
=
∑
cuts
 ∏
cut
momenta
∫
P
 ∏
subdiagrams
(−i) Disc
internal &
cut lines
[
i ψ(subdiagram)
]
, (4.1)
where D is some diagram that is reduced to a number of subdiagrams by cutting one or
more internal lines in all possible ways. Notice that in all cases the arguments of Disc, i.e.
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the energies that are not analytically continued, are all the internal lines plus whatever
external line resulted from a cut.
We will now make (4.1) more mathematically precise. The general cutting rules may
be stated in two steps: the first is diagrammatic (how to draw all “cut” diagrams), and the
second is algebraic (how to evaluate each of the cut diagrams).
Step 1. We begin with a connected diagram, D, which can be translated using the Feynman
rules of section 2 into a contribution ψ(D) to a wavefunction coefficient. We denote
by I the set of all internal lines in D (of which we are going to extract the imaginary
parts), and represent the appropriate Disc
I
[
iψ(D)
]
by highlighting the internal lines
(“I” stands for “Internal”). Each of these internal lines can be “cut” by replacing
them in D with a pair of external lines — i.e. if a line connecting vertices at t1 and
t2 is cut, then it is replaced by two external lines that connect t1 to the boundary
and t2 to the boundary. By cutting one or more of the highlighted lines, we produce
from the original diagram D a number of “cut diagrams”, which we denote by DC ,
where C ⊆ I is a list of which internal lines have been cut (“C” stands for “cut”
and all cut lines are highlighted). Notice that, as a result of the cutting, DC may
no longer be connected — we denote by D(n)C the connected subdiagrams contained
within DC , and furthermore use In ⊆ I to denote which internal lines are contained
in D(n)C .
Step 2. To each cut diagram DC , we associate a function D˜C [ψ] of the wavefunction coef-
ficients in the following way. First, notice that using the rules of section 2, we can
associate a ψ(D
(n)
C ) to each connected subdiagram D(n)C . We then take its Disc with
respect to both its internal lines In as well as any cut lines. Finally, we replace
each cut momenta pa listed in C with a pair of momenta {qa, q′a} and a factor of
the power spectrum, Pqaq′a , on the boundary. In formulae, this becomes:
DC = ∪nD(n)C ⇒ D˜C [ψ] ≡
 |C|∏
cut lines
a∈C
∫
qa q′a
Pqaq′a
 ∏
connected
subdiagrams
n
(−i) Disc
In{qa}
[
i ψ
(
D
(n)
C
)]
(4.2)
The general cutting rule then takes the simple form,
2|I|∑
cuts
C⊆I
D˜C [ψ] = 0 , (4.3)
where the sum is over all possible ways to cut the internal lines I in the diagram D. In par-
ticular, since the term C = {} corresponds to not performing any cuts, the corresponding
D{} is simply the original diagram D, and so separating this term out we have,
iDisc
I
[
i ψ(D)
]
=
2|I|−1∑
C⊆I
C 6={}
 |C|∏
a∈C
∫
qa q′a
Pqaq′a
∏
n
(−i) Disc
In{qa}
[
i ψ
(
D
(n)
C
)]
, (4.4)
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which expresses a particular discontinuity of the diagram D in terms of a sum over diagrams
that have at least one line cut. This is a more precise statement of the general cutting
relations described in words in (4.1), and is the central result of this work. This result
relates the discontinuity of an arbitrary diagram to those of diagrams with fewer loops
and/or fewer external legs. We will now prove (4.4). First, as a lemma we will prove
an algebraic identity for the imaginary part of the product of bulk-to-bulk propagators.
Second, we will integrate this identity to arrive at (4.4).
4.1 Lemma: a propagator identity
Our overall strategy is to first consider the integrands that appear in each wavefunction
coefficient. To each diagram D we associate an integrand Dˆ using the Feynman rules of
section 2, namely a product of bulk-to-boundary and bulk-to-bulk propagators. Since any
lines which are not highlighted can be factored out of the sum in (4.3), we need only focus
on the highlighted lines. The cutting procedure described above corresponds to replacing
the Gp(t1, t2) from each cut line with,
pa line cut ⇒ Gpa(t, t′)→ −2PpaKpa(t)Kpa(t′) , (4.5)
where the cut propagator factorises into separate functions of t and t′. IfDC is disconnected
by the cuts, then DˆC is defined analogously to (4.2): by taking the product of the imaginary
part of each connected subdiagram, after multiplying each by a factor9 of (2i)Ln , where Ln
is the number of loops in the subdiagram Dˆ(n)C ,
DC = ∪nD(n)C ⇒ DˆC ≡
∏
connected
subdiagrams,n
Im
[
(2i)LnDˆ(n)C
]
= 0 . (4.6)
We will now prove the following lemma: for any fixed ordering of the vertex times, (4.3)
is obeyed by the integrands, namely,
2|I|∑
cuts
C⊆I
DˆC = 0 . (4.7)
Looking ahead, in section 4.2 we will integrate this lemma over all times and loop momenta
to replace each DˆC with D˜C [ψ], which will hence prove (4.3).
Proof. We begin our proof of (4.7) by noting that there is always a largest time vertex
in the diagram D, which we denote by t¯ (we assume this is unique, but the same argu-
ment works if there are multiple vertices at this largest time). Bulk-to-bulk propagators
connected to the largest time vertex simplify because by the definition of Gp we have
Gp(t¯, t) = 2PpKp(t) ImKp(t¯) when t¯ ≥ t . (4.8)
9We will see below that this factor arises both because the Disc in (3.3) is related to Im by a factor of
2i, and also due to the overall factor of i1−L in the Feynman rules.
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! "##!
"##!! " "#$ "#$#!
$ "#
$#!
$ "##$% #!$#!
%$%& "#'%$&#!'
"# #! ! "$ "##!#$$%
! "# #!
$ "##!
"# #!
$ "##!
%$%& "#'%$&#!'
Figure 3. An internal line between the largest time vertex t¯ and another time tj is either (i)
connecting two otherwise disconnected components, or (ii) forming part of a loop (such that the
graph remains connected once it is removed). The pairwise additions shown correspond to (4.9)
and (4.9) respectively.
Then, by grouping the terms in (4.7) into pairs of cut diagrams which differ by the cutting
of only a single line which is connected to t¯, we can systematically reduce the number of
highlighted lines left to consider. For instance, consider two diagrams which differ only in
whether the highlighted line between t¯ and some other tj is cut. There are only two distinct
possibilities: either (i) cutting the tj → t¯ line separates the diagram into two disconnected
pieces, or (ii) the line tj → t¯ is part of a loop and so cutting this line does not separate
the diagram (but does reduce the number of loops by one). These two cases are shown in
figure 3. Considering each in turn:
(i) If the two disconnected subdiagrams after the cut are Rt¯ and Rtj , then by using (4.8)
we find
Im
[
Rt¯Gp(t¯, tj)Rtj
]
− 2PpIm
[
Rt¯Kp(t¯)
]
Im
[
Kp(tj)Rtj
]
= −2PpIm
[
Kp(t¯)∗Kp(tj)Rtj
]
Im [Rt¯] . (4.9)
Hence, we can treat this as an amputation of everything which was connected to
the t¯ vertex by the tj → t¯ line. Since the tj dependence of the right-hand-side has
completely factorised, we have extracted the Im [Rt¯] of the subdiagram containing t¯.
This reduces the number of highlighted lines we need to consider by the number of
highlighted lines in the amputated Rtj .
(ii) If instead the line tj → t¯ is part of a loop, then if we denote the connected remainder
after its removal by Rt¯tj , we have that (again using (4.8))
Im
[
2iGp(t¯, tj)Rt¯tj
]
− Im
[
2PpKp(t¯)Kp(tj)Rt¯tj
]
= −2PpIm
[
Rt¯tjKp(t¯)
∗Kp(tj)
]
,
(4.10)
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where there is an additional 2i in the first term since before the line is cut there is one
additional loop. This pairwise sum has reduced the number of internal lines remaining
by 1, and simply rescales the remaining diagram by a factor of −2PpKp(t¯)∗Kp(tj).
In either case, the line connecting t¯ and tj has been removed by this pairwise com-
bination and the number of highlighted lines left to consider has decreased. Repeating
this for all other highlighted lines which are connected to t¯ eventually amputates every
highlighted line, leaving a remainder (Rt¯ = 1) with vanishing discontinuity (Im [Rt¯] = 0
in (4.9)). This proves the claim of lemma (4.7). To make this more explicit, we provide
two simple examples below.
A tree-level example. Consider the simple tree-level diagram in which two vertices (at
times t1 and t2) are attached by highlighted lines to the largest time vertex t¯ (> t1 and
t2). Focusing on just these two lines, there are four distinct cuts which contribute to the
cutting rule (4.3). They can be collected into two pairs, as shown in figure 4,
Dˆ{} + Dˆ{p1} = Im
[
Rt1 Gp1(t1, t¯)Gp2(t2, t¯)Rt2
]
− 2Pp1Im [Rt1Kp1(t1)] Im
[
Kp1(t¯)Gp2(t2, t¯)Rt2
]
∝ Im [Gp2(t2, t¯)Rt2] (4.11)
Dˆ{p2} + Dˆ{p1,p2} = −2Pp2Im
[
Rt1 Gp1(t1, t¯)Kp2(t¯)
]
Im [Kp2(t2)Rt2 ]
+ 4Pp1Pp2Im
[
Kp1(t¯)Kp2(t¯)
]
Im [Rt1Kp1(t1)] Im [Rt2Kp2(t2)]
∝ −2Pp2Im
[
Kp2(t¯)
]
Im [Kp2(t2)Rt2 ] (4.12)
where the common constant of proportionality is −2Pp1Im
[
Kp1(t1)K∗p1(t¯)Rt1
]
, which is
easily confirmed using (4.8). The right-hand-sides of the above equations can be recognised
as the discontinuity of diagrams in which n = 1 vertices are attached to t¯, and indeed
their sum exactly cancels again by use of (4.8). Once these integrands are integrated
over all times and momenta to make full wavefunction coefficients, this relation effectively
reproduces the cutting rule (3.11) given in section 3.2.
A one-loop example. Consider the one-loop diagram shown in figure 5. Algebraically,
the cutting rules associate to each of these diagrams,
Dˆ{} + Dˆ{p1} = Im
[
(2i)Gp1(t¯, t1)Gp2(t¯, t1)
]− 2Pp1Im [Kp1(t¯)Kp1(t1)Gp2(t¯, t1)]
= −2Pp1Im
[
K∗p1(t¯)Kp1(t1)Gp2(t¯, t1)
]
(4.13)
Dˆ{p2} + Dˆ{p1,p2} = −2Pp2Im
[
Gp1(t¯, t1)Kp2(t¯)Kp2(t1))
]
+ 4Pp1Pp2Im
[
Kp1(t¯)Kp2(t¯)
]
Im [Kp1(t1)Kp2(t1)]
= +4Pp1Pp2Im
[
Kp2(t¯)
]
Im
[
K∗p1(t¯)Kp1(t1)Kp2(t1)
]
. (4.14)
The two terms on the right-hand side sum up to zero by virtue of (4.8). Once integrated
over all times and momenta, this reproduces the cutting rule (3.19) given in section 3.3.
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!"
"! ! "#"!#"" "#
!"
"! #"!#""
"# !""##"!
!"
"! ! "#"!#"" "#
!"
"! #"!#""
"# !""##"!
!""##"!
!""##"!
! # $
Figure 4. There are four ways to cut two propagators attached to the largest time vertex, t¯.
They can be paired together as shown in the first and second lines, which amputates all of the t1
dependence. The constants of proportionality are the same, and so adding the two diagrams as on
the third line shows that this sum vanishes (see (4.11) and (4.12)).
!"
"! ! "#! !""!
! "
#"
!"
"!#! #" #"
!"
"!#! #"
!"
"!#! #"
!"
"! #"
#!
#!
#
#
$#!$ % !"&$#!%"!&
$#!$ % !"&$#!%"!&
Figure 5. Diagrammatic representation of (4.13) and (4.14), showing the pairwise sum of two loop
diagrams that differ only by the cut of a single line. The two terms on the right-hand-sides exactly
cancel.
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4.2 Proof of the cutting rules
The lemma (4.7) generalises the identities (3.2) for ImG (used to cut a single propaga-
tor), (3.10) for ImG1G2 (used to cut two propagators at tree-level) and (3.13) or (3.18)
for ReG or ReG1G2 (used to cut one or two propagators in a loop) to any number of
propagators which form any number of loops. For convenience we list the first several of
these identities in appendix B. We will now use these general propagator identities to prove
the cutting rules (4.4) for an arbitrary L-loop diagram.
First, we can express an arbitrary wavefunction coefficient in terms of an integrand by
stripping off all external legs and their associated time integrals, as well as the momentum-
conserving δ-functions at each vertex,
ψ(D) =
 N∏
j=1
∫
dtj
∏
ka
Kka(tj)
 ψˆ(D)(t1, . . . , tN )× (δ functions) . (4.15)
For the original connected diagram, the integrand ψˆ contains a product of |I| internal
propagators (where |I| is the number of elements in the set I), an integral over their
momenta (all but L of these integrals may be fixed by the δ-functions), and an overall
factor of i1−L, as per the Feynman rules of section 2,
ψˆ(D)(t1, . . . , tN ) = i1−L
∫
p1...p|I|
Gp1 . . . Gp|I| (4.16)
where the propagators may depend on any of the times (t1, . . . , tN ). Using a Disc to take
the imaginary part of this product of propagators, we have,
−iDisc
I
[
i ψˆ(D)(t1, . . . , tN )
]
=
∫
p1...p|I|
(−2)1−LDˆ{} . (4.17)
For the cut diagrams DC , the integrand ψˆ(DC) is given by the analogue of (4.16) with the
cut propagators replaced as in (4.5). For instance, if after cutting the line p1 the diagram
remains connected (case (ii) above), then D{p1} has L−1 loops and two additional external
legs, and so Dˆ{p1} is related to a wavefunction integrand by,∫
q1q′1
Pq1q′1(−i)DiscIq1q′1
[
i ψˆ(D{p1})(t1, . . . , tN )
]
=
∫
p1...p|I|
(−2)1−LDˆ{p1} . (4.18)
On the other hand, if after cutting the line p1 the diagram becomes disconnected (case (i)
above), then we have,∫
q1q′1
Pq1q′1(−i)DiscI1q1
[
i ψˆ
(D(1){p1})(t1, . . . , tN )
]
(−i)Disc
I2q′1
[
i ψˆ
(D(2){p1})(t1, . . . , tN )
]
=
∫
p1...p|I|
(−2)1−LDˆ{p1} . (4.19)
Proceeding in this way for diagrams with two, three, . . . etc. cuts, we can replace each DˆC
in lemma (4.7) with products of ψˆ(DC) discontinuities,∫
p1...p|I|
(−2)1−L
2|I|∑
C⊆I
DˆC =
2|I|∑
C⊆I
 |C|∏
a∈C
∫
qa q′a
Pqaq′a
∏
n
(−i) Disc
In{qa}
[
i ψˆ
(
D
(n)
C
)
(t1, . . . , tN )
]
.
(4.20)
By our propagator lemma (4.7), the sum on the left-hand-side vanishes.
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The final step is then to multiply by the external propagators and perform the integrals
in (4.15) over the vertices (t1, . . . , tN ) at which they could be attached to the diagram. The
crucial property we adopt in this final step is that we can bring the time integrals and the
factors of Kk in (4.15) inside the argument of the Disc. This is allowed whenever there
exists an analytic continuation k¯ such that Kk(η) = K ∗¯k(η) for every η, since then,
Disc
[
R
n∏
a
Kka
]
≡ R
n∏
a
Kka +
(
R
n∏
a
Kk¯a
)∗
(4.21)
=
n∏
a
KkaDisc [R] , (4.22)
for any R, as discussed in [6, 7]. It is not always possible a priori to find such a k¯, but
a simple solution for k¯ turns out to exists under surprisingly general circumstances [5].
To see this, note that in Minkowski, where K ∼ eikt, the above implicit equation for
k¯ has solution k¯ = −k∗, reducing to simply a minus sign for real k. In analogy with
amplitudes, one can name this property Hermitian analyticity, namely Kk(η) = K∗−k∗(η).
The choice of a Bunch-Davies vacuum enforces Kk on any FLRW spacetime to match the
Minkowski result at early times. Then one can prove that, as long as the coefficients of
the linearized equations of motion are not singular in the past, Hermitian analyticity is
maintained as time evolves and in particular it remains valid even when the mode function
become dramatically different from those in flat spacetime [5]. Indeed, it is easy to see that
Hermitian analyticity is satisfied for both massless and conformally coupled scalar fields10
Kk(η) = (1− ikη)eikη , (massless scalar) (4.23)
Kk(η) =
η
η0
eikη , (conformally coupled scalar) . (4.24)
The above discussion allows us to promote each ψˆ(D) in (4.20) to ψ(D), and hence proves
the general cutting rule (4.4).
A one-loop example. For instance, for the one-loop example given above (see figure 5),
the diagram with zero-, one- or two-cuts corresponds to wavefunction coefficient integrands,
ψˆ
(D)
{k}(t1, t2) =
∫
p1p2
Gp1(t1, t2)Gp2(t1, t2) ,
ψˆ
(D{p1})
{k}q1q′1 (t1, t2) = i
∫
p2
Kq1(t1)Kq′1(t2)Gp2(t1, t2) ,
ψˆ
(
D
(1)
{p1,p2}
)
{k}q1q2 (t1, t2) = iKq1(t1)Kq2(t1) , ψˆ
(D(2){p1,p2})
{k}q′1q′2 (t1, t2) = iKq′1(t2)Kq′2(t2) . (4.25)
10Notice that massless gravitons have the same mode functions as massless scalars, and so they too obey
Harmitian analyticity. Also, where the limit is finite we have taken η0 → 0.
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We can therefore write the DˆC given in equations (4.13) and (4.14) above in terms of the
wavefunction integrands,∫
p1p2
Dˆ{} =
∫
p1p2
Im [2iGp1(t1, t2)Gp2(t1, t2)]
= −iDisc
[
iψˆ
(D)
{k}(t1, t2)
]
(4.26)∫
p1p2
Dˆ{p1} =
∫
p1p2
−2Pp1Im
[
Kp1(t¯)Kp1(t1)Gp2(t¯, t1)
]
=
∫
q1q′1
−Pq1q′1iDiscq1q2
[
iψˆ
(D{p1})
{k}q1q′1(t1, t2)
]
(4.27)
∫
p1p2
Dˆ{p1,p2} =
∫
p1p2
4Pp1Pp2Im [Kp1(t1)Kp2(t1)] Im [Kp1(t2)Kp2(t2)]
=
∫
q1q′1
q2q′2
−Pq1q′1Pq2q′2Discq1q2
[
iψˆ
(D(1){p1,p2})
{k}q1q2 (t1, t2)
]
Disc
q′1q
′
2
[
iψˆ
(D(2){p1,p2})
{k}q′1q′2 (t1, t2)
]
.
(4.28)
The propagator lemma (4.7) for this diagram can therefore be written as,
0 =
∫
p1p2
(
Dˆ{} + Dˆ{p1} + Dˆ{p2} + Dˆ{p1,p2}
)
= −iDisc
[
iψˆ
(D)
{k}(t1, t2)
]
+
∫
q1q′1
Pq1q′1(−i)Discq1q2
[
iψˆ
(D{p1})
{k}q1q′1(t1, t2)
]
+
∫
q2q′2
Pq2q′2(−i)Discq1q2
[
iψˆ
(D{p1})
{k}q1q′1(t1, t2)
]
+
∫
q1q′1
q2q′2
Pq1q′1Pq2q′2(−i)Discq1q2
[
iψˆ
(D(1){p1,p2})
{k}q1q2 (t1, t2)
]
(−i)Disc
q′1q
′
2
[
iψˆ
(D(2){p1,p2})
{k}q′1q′2 (t1, t2)
]
.
(4.29)
Finally, multiplying by the external propagators and performing the integrals over the times
(t1, t2) replaces each of these integrands ψˆ(D) with the corresponding coefficient ψ(D), and
therefore (4.29) implies the cutting rule,
iDisc
[
iψ(D)
]
=
3∑
C⊆{p1,p2}
C 6={}
 |C|∏
a∈C
∫
qa q′a
Pqaq′a
∏
n
(−i)Disc
{qa}
[
i ψ
(
D
(n)
C
)]
, (4.30)
for this diagram, where in this case the internal momenta are integrated over so the Disc
with no argument on the left-hand-side corresponds to analytically continuing all (and
only) the external momenta.
4.3 Extension to multiple fields of any mass and spin
The Cosmological Cutting Rules have been presented so far for a single massless scalar
field in de Sitter spacetime with a Bunch-Davies vacuum. However, the same rules apply
to the much more general case of any (finite) number of fields of any mass and spin. Here,
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we only sketch the main argument and refer the reader to [5] for more details on spinning
fields and more general FLRW spacetimes.
Our proof so far relied on two properties. The first are the propagators identities
proven in the lemma in section 4.1. These are very general and only rely on the form of the
bulk-to-bulk propagator Gp in terms of the bulk-to-boundary propagator Kk. The proof
assumes nothing about the function Kk. This result is therefore valid for any number of
fields with any mode functions. It is straightforward to extend this proof to allow for fields
of different species/spins. This amounts to decorating the propagators in the lemma (4.7)
with additional indices that denote any additional quantum numbers. For example, for the
cutting rule (3.12), this amounts to writing,
Disc
pLpR
[
iψ
{αL}{αM}{αR}
{kL}{kM}{kR}
]
=
∫
qLq′L
Disc
qL
[
iψ
{αL}βL
{kL}qL
]
P βLqLq′L
Disc
q′LpR
[
iψ
βL{αM}{αR}
q′L{kM}{kR}
]
+
∫
qRq′R
Disc
pLqR
[
iψ
{αL}{αM}βR
{kL}{kM}qR
]
P βRqRq′R
Disc
q′R
[
iψ
βR{αR}
q′R{kR}
]
+
∫
qLq′L
qRq′R
Disc
qL
[
iψ
{αL}βL
{kL}qL
]
P βLqLq′L
Disc
q′LqR
[
iψ
βL{αM}βR
q′L{kM}qR
]
P βRqRq′R
Disc
q′R
[
iψ
βR{αR}
q′R{kR}
]
(4.31)
where the indices α and β collect the other quantum numbers of the fields, such as field
type (e.g. flavor), helicity, charges and so on. Notice that these indices are always paired
up with the associated momenta. We can therefore omit to write them altogether if we
improve our notation to include these indices inside the various k’s, p’s and q’s. The
integrals over q’s should then be interpreted as having an additional sum over the relevant
quantum numbers, for example all the possible helicity of a given spinning field. We refer
the reader to [5] for a more explicit discussion and notation.
The second property we needed to translate the propagator identities into equations
for the wavefunction coefficient, is that we can find a k¯ such that K ∗¯
k
(η) = Kk(η) for all
times η. When the fields obey the Bunch-Davies vacuum, this condition is satisfied by
k¯ = −k∗, and we refer to this property of Kk as Hermitian analyticity. In [5] we prove
that Hermitian analyticity is valid for fields of any mass and spin on any FLRW spacetime,
provided that a weak technical assumption is satisfied by the coefficients of the linearized
equations of motion.
5 Inferring loops from trees using perturbative unitarity
The general cutting rule derived above allows us to compute the Disc of a loop-level wave-
function coefficient in terms of simpler tree-level coefficients. In this section, for a variety
of interactions on both Minkowski and de Sitter spacetime backgrounds we will show ex-
plicitly how the cutting rules can be used to infer the 1-loop Disc of the Gaussian width,
ψk1k2 . This provides a new way of estimating when perturbative unitarity breaks down
from a purely tree-level calculation. Finally, in section 5.4 we relate these results to the
power spectrum.
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Perturbative unitarity. Unitarity can be used to place a lower bound on the size of
loops, given specified tree-level contributions. This is familiar from scattering amplitudes on
flat space, where the perturbative optical theorem constrains ImA1-loop ≥ ∫Momenta |Atree|2.
This can be used to determine at what scale perturbation theory breaks down, in particular
when
∫
Momenta |Atree|2 ≥ |Atree| it signals that |A1-loop| must be larger than |Atree| if the
theory is to remain unitary. Our goal in this section is to apply the above cutting rules
in a similar spirit, using them to infer the size of 1-loop corrections from purely tree-level
calculations.
Momenta integrals. In the explicit examples below, we will need to evaluate momentum
integrals of the form
∫
q1q2 δ
3(q1 +q2−k) which appear in the cutting rules. Since we have
implicitly assumed throughout that the Disc operation commutes with such integrals,
Disc
[∫
q1q2
δ3(q1 + q2 − k) f(k, q1, q2)
]
=
∫
q1q2
δ3(q1 + q2 − k) Disc
q1q2
[f(k, q1, q2)] , (5.1)
we must take care to adopt integration variables which are suitably invariant under k → −k.
For example, one possible choice of integration variables is,∫
d3q1d3q2 δ3(q1 + q2 − k) f(k, q1, q2) = 2pi
∫ ∞
0
dq1
∫ |q1+k|
|q1−k|
dq2
k
q1q2 f(k, q1, q2) , (5.2)
where the integration limits11 are such that the Disc
[∫ |q1+k|
|q1−k|
dq2
k
]
= 0, and so taking Disc of
this integral amounts to integrating Disc
q1q2
[f(k, q1, q2)], as required. To simplify the algebra,
we will also make use of the following trick: for integrands f(k, q1, q2) with the property12
that f(−k,−q1,−q2) = −f∗(k, q1, q2), we can write the Disc of (5.2) as,
Disc
[∫
d3q1d3q2 δ3(q1 + q2 − k) f(k, q1, q2)
]
= pi
∫ ∞
0
dq+
∫ k
−k
dq−
k
q1q2 Disc
q1q2
[f(k, q1, q2)] ,
(5.3)
where q+ = q1 + q2 and q− = q1− q2. (5.3) is often easier to perform since the two integrals
are now independent.
All momentum integrals are to be computed with the prescription k → k − i (i.e. k
has a small negative imaginary part) to move poles from the real axis. For instance, using
the fact that, ∫ ∞
0
dq
q
qn
(q + k)r =
Γ (n) Γ (r − n)
Γ(r) k
n−r , (5.4)
for all complex k, the difference of two such integrals at k−i and e−ipi(k+i) corresponds to,
lim
→0+
∫ ∞
0
dq
q
qn
(q2 + e−ipik2 − i)r =
Γ
(
n
2
)
Γ
(
r − n2
)
2Γ(r) (−ik)
n−2r , (5.5)
11Note that the limits for q2 follow from q22 = |k − q1|2, and it is important to keep the modulus on the
q1 + k upper limit since we allow for k < 0 when taking the Disc.
12In general, (5.2) corresponds to the integration range
∫∞
k
dq+
∫ +k
−k
dq−
k
, which no longer com-
mutes with the Disc operation. However the property f(−k,−q1,−q2) = −f∗(k, q1, q2) ensures that
Disc
[∫ k
0 dq+
∫ +k
−k
dq−
k
f(k, q1, q2)
]
= 0, which then allows us to write (5.3).
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where writing −k2 as e−ipik2 in the denominator ensures that we have the correct branch
of
√−k2 = −ik on the right-hand-side. The useful identities (5.4) and (5.5) will be used
several times below.
5.1 On Minkowski
For a massless scalar field φ on a Minkowski background, we use mode functions fk(t) =
e−ikt/
√
2k, and the corresponding power spectrum is, Pk = |fk|2 = 1/(2k). The bulk-to-
boundary and bulk-to-bulk propagators are given by,
Kk(t) = eikt , Gk(t1, t2) =
eikt2
k
sin(kt1)θ(t1 − t2) + (t1 ↔ t2) . (5.6)
We will use the cutting rules to compute the one-loop correction to the (Disc of the)
Gaussian width, ψk1k2 , from both a quartic φ4 interaction and a cubic φ3 interaction.
Crucially, while the full ψ1-loopk1k2 in both cases is divergent and requires renormalisation, the
Disc
[
i ψ1-loopk1k2
]
is finite and can be inferred directly from the tree-level non-Gaussianities
ψ4 and ψ3.
5.1.1 φ4 on Minkowski
For the interaction Lint = 14!λφ4, the tree-level quartic wavefunction coefficient is
ψtreek1k2k3k4 = λ
δ˜3 (k1 + k2 + k3 + k4)
k1 + k2 + k3 + k4
. (5.7)
This is the only simple input required for the cutting rule (3.15), which fixes the Disc of
the 1-loop ψk1k2 as,
iDisc
[
iψ1-loopk1k2
]
=
∫
qq′
Pqq′ (−i)Disc
qq′
[
iψtreek1k2qq′
]
. (5.8)
Explicitly, from (5.7) (and the definition (3.3) of Disc) we can straightforwardly compute
the integrand on the right-hand-side,
(−i)Disc
qq′
[
iψtreek1k2qq′
]
= λδ˜
3 (k1 + k2 + q + q′)
k1 + k2 + q + q′
+ λδ˜
3 (−k1 − k2 − q − q′)
−k1 − k2 + q + q′
= λ 2(q + q
′)
(q + q′)2 − (k1 + k2)2 δ˜
3 (k1 + k2 + q + q′) . (5.9)
Unlike the loop momentum integral required to evaluate ψ1-loopk1k2 explicitly, the integration
on the right-hand-side of (5.8) over external momenta is finite,∫
qq′
Pqq′(−i)Disc
qq′
[
iψtreek1k2qq′
]
= λ
pi2
∫ ∞
0
dq
q2
4q2 − (k1 + k2)2 δ˜
3 (k1 + k2)
= iλ16pi (k1 + k2) δ˜
3 (k1 + k2) , (5.10)
using the integral identity (5.5). This simple finite integral has computed for us the Disc
of the 1-loop quadratic coefficient,
Disc
[
i ψ1-loopk1k2
]
= λ16pi (k1 + k2) δ˜
3 (k1 + k2) . (5.11)
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We note that (5.11) is consistent with the “naive dimensional analysis” (NDA) power
counting typically employed for loop amplitudes on flat space13 [54], which keeps track
of powers of 4pi. In this case,14 NDA would give ψ1-loopk1k2 ∼ λ k/(4pi)2, and we expect
Disc
[
iψ1-loopk1k2
]
to contain an additional power of pi since it arises from a logarithmic branch
cut. The cutting rules can therefore be viewed as an efficient way of fixing the numerical
coefficients, as well as the precise dependence on the momenta, in these power counting
formulae. This extends a similar application of unitarity in [51], which focused on scattering
amplitudes in the subhorizon limit, to wavefunction coefficients.
Comparison with explicit computation. In this simple example, we can check the
result (5.11) by performing the loop integral explicitly. The quadratic coefficient (with
δ-function removed) is given up to O(λ) by,
ψ′k,−k = −k + λ
∫ 0
−∞
dtKk(t)Kk(t)
∫
p
Gp(t, t)
= −k + λ4k
∫
ddp
(2pi)d
1
k + p
= −k + λ4k
Sd−1
(2pi)d 2 Γ (1− d) k
d−1
= −k
[
1 + 2λ16pi2
( 1
d− 3 + log (k) + local
)]
, (5.12)
where Sn is the surface area of the unit n-sphere (i.e. S2 = 4pi) and we have used (5.4)
to evaluate the momentum integral. “Local” denotes finite terms which are analytic in k,
and are therefore sensitive to the renormalisation prescription (i.e. can be fixed by adding
local counterterms). In particular, these local terms are purely real.
The loop integral (5.12) contains a 1/(d− 3) divergence in dimensional regularisation,
but when we take the Disc we pick out only the (finite) coefficient of the log(k) running,
Disc
[
i ψ′k,−k
]
= −ik 2λ16pi2
(
log(k)− log
(
e−ipik
))
= λ16pi 2k (5.13)
which indeed agrees with our cutting rule (5.11) on the support of the δ-function.
Perturbative unitarity. Just as for scattering amplitudes on Minkowski, we can use this
Disc to place a bound on the size of the 1-loop correction. Comparing the tree-level result,
ψtree ′k,−k = −k, with the loop-level, Disc
[
ψ1-loop ′k,−k
]
, we have that |λ| . 8pi is necessary for
this interaction to respect unitarity perturbatively. More precisely, while ψ1-loop2 contains
local terms which can be freely fixed by imposing a renormalisation condition (the finite
terms in (5.12)), it also contains a non-local log(k) running, the coefficient of which is an
13Power counting schemes analogous to NDA were developed for inflation in [50, 51] (see also [52, 53] for
theories with small cs in particular).
14Note that while NDA was generalised to d dimensions in [55], this assumed d-dimensional Lorentz-
invariant kinetic terms for the fields — in our case, although the loop integrals are done over only spatial
momenta in d = 3, the underlying field theory is four dimensional, and so we retain the 4pi counting of
(3 + 1) dimensions.
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unambiguous prediction of the perturbative theory. Supposing that |ψ1-loop2 | is initially set
to be less than |ψtree2 | at some k∗, the condition that λ < 8pi ensures that |ψ1-loop2 | < |ψtree2 |
at scales within an order of magnitude of k∗. Interestingly, we note that λ < 8pi is the same
bound that one obtains from the 2→ 2 scattering amplitude (for which ImA1-loop2→2 = λ2/8pi
and Atree2→2 = λ).
5.1.2 φ3 on Minkowski
We will now consider a cubic potential interaction for a massless scalar field on Minkowski.
Although this potential is not bounded from below, it serves as a useful illustration of how
the cutting rules correctly reproduce the Disc of various 1-loop diagrams.
For the interaction Lint = 13!µφ3, the tree-level cubic wavefunction coefficient is,
ψtreek1k2k3 = µ
δ˜3 (k1 + k2 + k3)
k1 + k2 + k3
. (5.14)
There are also three tree-level exchange contributions to
ψk1k2k3k4 = ψ
(ps)
k1k2k3k4 + ψ
(pt)
k1k2k3k4 + ψ
(pu)
k1k2k3k4 , (5.15)
which are given by
ψ
(ps) ′
k1k2k3k4 =
µ2
(k1 + k2 + k3 + k4)(k1 + k2 + ps)(k3 + k4 + ps)
, (5.16)
plus the two permutations of the external legs. These are the only inputs needed to infer
the Disc
[
iψ1-loopk1k2
]
using the cutting rules. Explicitly, we require the following Disc of (5.14)
and (5.16),
(−i)Disc
q1q2
[
iψtreekq1q2
]
= µ 2q+
q2+ − k2
δ˜3(k1 + q1 + q2)
(−i)Disc
q p0
[
iψ
(p0) ′
k,−k,q,−q
]
= µ
2
8
( 1
kq(k + q) +
1
−kq(−k + q)
)
(5.17)
(−i)Disc
q p
[
iψ
(p) ′
k,q,−q,−k
]
= µ
2
2
( 1
(k + q)(k + q + p)2 +
1
(−k + q)(−k + q + p)2
)
where here p0 = 0, p is arbitrary and q+ = q1 + q2.
In this theory, there are two diagrams which contribute to ψ1-loopk1k2 ,
(a) (b)
which we label (a) and (b). Applying the cutting rules to diagram (a) we have,
iDisc
[
iψ
(a)
k1k2
]
=
∫
q2q′2
Pq2q′2(−i) Discq2q′2 p1
[
iψ
(p1)
k1q2q′2k2
]
+
∫
q1q′1
Pq1q′1(−i) Discq1q′1 p2
[
iψ
(p2)
k1q1q′1k2
]
+
∫
q1q′1
q2q′2
(−i)Disc
q1q2
[
iψtreek1q1q2
]
Pq1q′1Pq2q′2 (−i)Discq′1q′2
[
iψtreeq′1q′2k2
]
, (5.18)
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where p1 = |k1 +q2| and p2 = |k1 +q1|. Note that the exchange contributions on the first
line vanish identically,∫
q2q′2
Pq2q′2 (−i) Discq2q′2 p1
[
iψ
(p1)′
k,q2,q′2,−k
]
= 0 ,
∫
q1q′1
Pq1q′1 (−i) Discq1q′1 p2
[
iψ
(p2)′
k,q1,q′1,−k
]
= 0 ,
(5.19)
while the ψ3 × ψ3 contribution on the second line can be reduced to a single integral
using (5.3),
∫
q1q′1
q2q′2
(−i)Disc
q1q2
[
iψtreekq1q2
]
Pq1q′1Pq2q′2 (−i)Discq′1q′2
[
iψtreeq′1,q′2,−k
]
= µ
2
16pi2
∫ ∞
0
dq+
(
2q+
q2+ − k2
)2
(5.20)
which is given by the integral identity (5.4). So altogether, diagram (a) contributes to the
Gaussian width as,
Disc
[
iψ
(a)′
k,−k
]
= + µ
2
16pi
1
k
. (5.21)
Now applying the cutting rules to diagram (b) we have,
iDisc
[
iψ
(b)
k1k2
]
=
∫
qq′
Pqq′(−i)Disc
qq′ p0
[
iψ
(p0)
k1k2q1q2
]
, (5.22)
where we have used that Disc
q
[
iψtreek1k2q
]
= 0 (since k1 +k2 = 0 on imposing the δ-functions)
to discard the two diagrams in which the internal line with p = 0 is cut. This exchange
contribution can also be written in the form (5.4),
∫
qq′
Pqq′(−i)Disc
qq′ p0
[
iψ
(p0) ′
k,−k,q,q′
]
= µ
2
16pi2
∫ ∞
0
dq
1
q2(q2 − k2) (5.23)
and so diagram (b) contributes to the Gaussian width as,
Disc
[
iψ
(b) ′
k,−k
]
= − µ
2
16pi
1
2k . (5.24)
Altogether, unitarity requires that Disc
[
iψ1-loopk1k2
]
is given by the sum of (5.21) and (5.24),
Disc
[
iψ1-loop ′k,−k
]
= + µ
2
16pi
1
2k . (5.25)
We stress that this required only knowledge of the tree-level coefficients (5.14) and (5.16),
and each momenta integral that we encountered in the cutting rules was manifestly finite
(and did not require any regularisation or renormalisation). Before we move on to inflation-
ary wavefunction coefficients in the next subsection, let us briefly show how the Disc (5.21)
and (5.24) could have been computed by instead performing the explicit loop integral.
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Comparison with explicit computation. Note that diagram (a) can be computed
directly using,
ψ
(a) ′
k,−k = µ
2
∫ 0
−∞
dt1
∫ 0
−∞
dt2Kk(t1)Kk(t2)
∫
p1p2
Gp1(t1, t2)Gp2(t1, t2) δ˜3(p1 + p2 + k) ,
= µ
2
4
∫
p1p2
2k + p1 + p2
k(k + p1)(k + p2)(k + p1 + p2)2
δ˜3(p1 + p2 + k) . (5.26)
This integral can be performed using the d-dimensional version of (5.3) (which is given
in (C.6)) to replace ddp1ddp2 with dp+dp−. Carrying out the dp− integral leaves,
ψ
(a)′
k,−k =
µ2
16pi2
∫ ∞
k
dp+
p+ + 2k − 2(p+ + k) log
(
p++3k
p++k
)
k(p+ + k)2
(
pd−3+ +O(d− 3)
)
, (5.27)
where we have discarded terms suppressed by (d− 3) that do not contribute to the diver-
gence or the logarithmic running. In fact, the only term which diverges here is the integral,∫∞
k dp+ p
d−2
+ /(p+ + k)2, which gives,
ψ
(a)′
k,−k =
µ2
16pi2k
[ 1
3− d − log (k) + finite
]
, (5.28)
and the remaining finite part is purely real. The discontinuity again comes from the
logarithmic branch cut,
Disc
[
iψ
(a) ′
k,−k
]
= µ
2
16pi2
log(k)− log(e−ipik)
ik
= + µ
2
16pi
1
k
, (5.29)
and coincides with the result which we obtained from unitarity (5.25).
The tadpole diagram (b) is given by,
ψ
(b) ′
k,−k = µ
2
∫ 0
−∞
dt1
∫ 0
−∞
dt2Kk(t1)Kk(t1)G0(t1, t2)
∫
p
Gp(t2, t2)
= − µ
2
16pi2
1
2k
[ 1
3− d − log (k) + finite
]
, (5.30)
which matches the Disc inferred using unitarity,
Disc
[
iψ
(b)
k,−k
]
= + µ
2
16pi2
− log(k) + log(e−ipik)
2ik = −
µ2
16pi
1
2k . (5.31)
While using the unitarity cuts to compute diagrams (a) and (b) did not provide any
information about the divergent part of ψ1-loop, it directly provides the finite Disc (i.e. the
coefficient of the log(k) running) without the need for laborious loop integrals.
5.2 On de Sitter
For a massless scalar field φ on de Sitter, we use the Bunch-Davies mode function,
fk(η) =
H(1 + ikη)
k
e−ikη√
2k
, (5.32)
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and the corresponding (late time) power spectrum is, Pk = |fk|2 = H2/2k3. In this case,
the relevant bulk-to-boundary and bulk-to-bulk propagators are,
Kk(η) = (1− ikη) eikη , (5.33)
Gp(η1, η2) =
H2
p3
(1− ipη2) eipη2 [(sin(pη1)− pη1 cos(pη1)) θ(η1 − η2) + (η1 ↔ η2)] .
We will now use the Cosmological Cutting Rules to compute the one-loop correction to
Disc [iψk1k2 ] from the cubic vertices φ˙3 and φ˙(∂iφ)2. The one-loop correction from φ˙3 has
been computed previously (see e.g. [47]), however the unitarity derivation we present here
is significantly shorter and less laborious. To the best of our knowledge, the loop diagrams
containing φ˙(∂φ)2 vertices have not been computed before — likely due to their algebraic
complexity — and here we are able to find the Disc of these diagrams. Let us begin with
the φ˙3 interaction only, and then move on to include the φ˙(∂φ)2.
5.2.1 φ˙3 on de Sitter
For the interaction Lint = Cφ˙3(−Hη)−1(φ′)3, where φ′ = ∂ηφ in conformal time, the late-
time tree-level cubic wavefunction coefficient from the Bunch-Davies initial state is,
ψtree ′k1k2k3 = −
2Cφ˙3
H
k21k
2
2k
2
3
(k1 + k2 + k3)3
. (5.34)
There is also a quartic coefficient ψ4 sourced by the s, t or u-channel exchange of φ, but
these turn out not to contribute at this order.15
Again there are two diagrams which contribute to ψ1-loopk1k2 , which we label (a) and (b)
as above. The cutting rule (5.18) for Disc
[
iψ
(a)
k1k2
]
contains exchange terms but once again
these integrals vanish identically as in (5.19). The remaining ψ3 × ψ3 integral is given by,∫
q1q′1
q2q′2
(−i)Disc
q1q2
[
iψtreekq1q2
]
Pq1q′1Pq2q′2 (−i)Discq′1q′2
[
iψtree−kq′1q′2
]
=
H2C2
φ˙3
16pi2
∫ ∞
0
dq+
k4
(
q3+ + 3q+k2
)2 (15q4+ − 10q2+k2 + 3k4)
15
(
q2+ − k2
)6 , (5.35)
where we have changed to (q+, q−) variables using (5.3) and performed the q− integral.
The remaining q+ integral can be carried out using the identity (5.5), and consequently
the cutting rules determine the one-loop discontinuity to be,
Disc
[
iψ
(a)
k−k
]
= +
H2C2
φ˙3
16pi k
3 2
15 . (5.36)
For the one-loop diagram (b), once again we have that Disc
q
[
iψk1k2q
]
= 0 and so the
cutting rule is again simply (5.22). Unlike for φ3 Minkowski space, for φ˙3 on de Sitter this
15This can be seen in the following way. At large q, the quartic coefficient scales as, ψk−kq−q ∼ k4/q, in
all three channels. The integrand Pqψk−kq−q ∼ k4/q4 and therefore
∫
qq′ Pqq′ψk1k2qq′ does not diverge in
d = 3 dimensions — consequently there is no logarithmic dependence on k, and so the Disc of this integral
vanishes.
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contribution vanishes because the exchange contribution vanishes,
Disc
[
iψ
(b)
k1k2
]
= 0 , (5.37)
and so the entire discontinuity in ψk1k2 at one-loop is given by (5.36).
5.2.2 φ˙(∂iφ)2 on de Sitter
Now we consider the general cubic interaction,
Lint = −
Cφ˙3
Hη
(
φ′
)3 − Cφ˙(∂φ)2
Hη
φ′(∂iφ)2 , (5.38)
which contains both φ˙3 and φ˙(∂φ)2 interactions. The corresponding cubic wavefunction
coefficient can be found in [56],
ψtreek1k2k3 =
1
2Hk3T
[
Cφ˙3 24e
2
3 + Cφ˙(∂φ)2
(
24e23 − 8kT e2e3 − 8k2T e22 + 22k3T e3 − 6k4T e2 + 2k6T
)]
(5.39)
where
e3 = k1k2k3 , e2 = k1k2 + k1k3 + k2k3 kT = k1 + k2 + k3 . (5.40)
The exchange contribution to ψ4 is schematically,16
ψ′k,−k,q,−q ∼
∫
dη
η
ηd ψ′k,q,−k−qψ
′
−k,−q,k+q (5.41)
and so since17 ψ′k,q,−k−q ∼ Cφ˙3q1 +Cφ˙(∂φ)2q1 at large q, we expect ψ′k,−k,q,−q ∼ q−1. This
expectation indeed matches the explicit computation. The integral
∫
qq′ Pqq′ψk1k2qq′ there-
fore does not contain any divergence and so drops out of the cutting rules, and therefore
only the twice-cut diagram contributes. Unitarity therefore fixes the 1-loop Disc to be,
iDisc
[
iψ
(a) ′
k1k2
]
=
∫
q1q′1
q2q′2
(−i)Disc
q1q2
[
iψtreekq1q2
]
Pq1q′1Pq2q′2 (−i)Discq′1q′2
[
iψtree−kq′1q′2
]
= H
2
pi
ik3
[ 3
10C
2
φ˙3
− 910Cφ˙3Cφ˙(∂φ)2 +
51
20C
2
φ˙(∂φ)2
]
. (5.42)
Although the overall factors of H, k and pi could have been inferred from power counting,
the numerical coefficients in (5.42) could not have been. The cutting rules are therefore
providing an efficient route to this part (the Disc) of the 1-loop wavefunction, completely
removing the need for regularising and performing complicated loop integrals.
16Formally, there is an additional contribution with integrand ψk,−k,0ψq,−q,0, but this vanishes because
pi is derivatively coupled.
17Although the Cφ˙(∂φ)2 term in ψ3 seems to ∼ q3, the numerical coefficients are such that it only ∼ q at
large q, a consequence of the soft theorem for the squeezed bispectrum [56].
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5.3 For the EFT of inflation
Finally, let’s consider inflation. Following the EFT approach of [21], we consider the low-
energy effective action for perturbations about an expanding FLRW background. Although
this background introduces an explicit time dependence, temporal diffeomorphisms can be
restored (non-linearly realised) by introducing a single scalar degree of freedom, pi, which
decouples from the metric perturbations in the so-called decoupling limit (MP →∞ with
fpi fixed). In this decoupling limit, the scalar perturbations in the EFT of Inflation are
described by,
S[pi] =
∫
dtd3x a3 f
4
pi
c3s
[
p˙i2
2 − c
2
s
(∂ipi)2
2a2 + Cp˙i3 p˙i
3 + Cp˙i(∂pi)2 p˙i
(∂ipi)2
a2
]
, (5.43)
where f4pi = 2csM2P |H˙| ≈ (60H)4 is the energy scale associated with the symmetry break-
ing (fixed by the power spectrum) and cs is the sound speed. The non-linearly realised
symmetry fixes Cp˙i3 and Cp˙i(∂ipi)2 in terms of cs and one additional Wilson coefficient, con-
ventionally denoted by c˜3 [57],
Cp˙i3 =
1
2
(
1− c2s
)(
1 + 2c˜33c2s
)
, Cp˙i(∂ipi)2 = −
1
2
(
1− c2s
)
. (5.44)
cs and c˜3 are constrained by the primordial bispectrum [58].
5.3.1 Wavefunction at one-loop
We will now use the Cosmological Cutting Rules to compute ψ1-loop from the EFT of Infla-
tion (5.43). Unlike the φ of previous subsections, the kinetic term for pi is not canonically
normalised — this is accounted for via the rescaling pi = φ/f2pi and x = csx˜. Note that
when considering an approximately de Sitter spacetime background, we can write (5.43)
in terms of conformal time Ht = − log(−Hη) as,
S[φ] =
∫
dηd3x˜
[ 1
2H2η2
[
(φ′)2 − (∂˜iφ)2
]
− Cp˙i3
f2piHη
(φ′)3 − Cp˙i(∂pi)2
f2pic
2
sHη
φ′(∂˜iφ)2
]
, (5.45)
where a prime denotes derivatives with respect to conformal time. This expression has the
same form as the interactions (5.38) considered above, with coefficients,
Cφ˙3 =
Cp˙i3
f2pi
and Cφ˙(∂φ)2 =
Cp˙i(∂pi)2
f2pic
2
s
. (5.46)
Since we have massaged (5.43) into the same form as (5.38), we can follow the same
steps described in section 5.2 to arrive at the (Disc of the) one-loop coefficient ψk1k2 of18
φk1φk2 ,
iDisc
[
iψ1-loopk1k2
]
= H
2
f4pi
ik3
480pi
(1− c2s)2
c4s
[
(4c˜3 + 9 + 6c2s)2 + 152
]
(5.47)
=: H
2
f4pi
ik3
16pi γ(cs, c˜s) .
18In other words, all the wavefunction coefficients we quote refer to the canonically normalized field.
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Again we note that, while the overall scaling of this quantity could have been inferred from
dimensional analysis alone, the cutting rules have allowed us to go much further by also
providing the precise form of the coefficient γ(cs, c˜3). While the C2p˙i3 contribution to γ(cs, c˜3)
could have been extracted from the explicit one-loop computation performed in [47], we
are not aware of any previous computation of this general expression (which, without the
cutting rules, would require performing the explicit loop integrals with p˙i(∂ipi)2 vertices).
Quartic interactions. Note that although we only considered the leading cubic interac-
tions in (5.43), our results are robust against a potentially large quartic corrections. This
is since the one-loop correction from p˙i4 does not diverge, as noted previously in [47], and
consequently it does not affect the Disc [iψk1k2 ] at one-loop. We can confirm this straight-
forwardly using the cutting rules. For the interaction19 14!Cp˙i4(pi′)4, the late-time tree-level
quartic wavefunction coefficient from the Bunch-Davies initial state is,
ψtree ′k1k2k3k4 = −Cp˙i4
k21k
2
2k
2
3k
2
4
(k1 + k2 + k3 + k4)5
. (5.48)
Using the cutting rule (3.15), the corresponding contribution to Disc [iψk1k2 ] is given by
the integral,∫
qq′
Pqq′ (−i)Disc
qq′
[
iψk,−k,q,q′
]
= Cp˙i416pi2
k4
4
∫ ∞
0
dq q3
(q2 − k2)5
(
q5 + 10k2q3 + 5k4q
)
, (5.49)
which vanishes once evaluated using the integral identity (5.5). This shows that the Disc
of the 1-loop Gaussian width is insensitive to the tree-level trispectrum, at least in the
limit where p˙i4 dominates over p˙i2(∂ipi)2 and (∂ipi)4 (which is natural since the latter two
interactions are fixed in terms of cs by the non-linearly realised symmetry).
5.4 Physical interpretation
Let us now interpret the physical meaning of the discontinuity (5.47) in ψ1-loop2 .
Source of the Disc. In the Minkowski examples in section 5.1, we found by explicit com-
putation that the Disc corresponded to the coefficient of the log(k) running of ψ1-loop. This
is exactly analogous to the logarithmic discontinuities encountered in flat space scattering
amplitudes at one-loop. It is therefore tempting to conclude from (5.47) that,
ψ1-loopk1k2
?= H
2
f4pi
k3
16pi2 [−γ log (k) + divergence + local] (5.50)
where the “local” remainder is a real analytic function of k. This is indeed the form of
the one-loop corrections found in Weinberg’s original article [20] (see also [59, 60]), in
which the loop integrals were computed using a certain form of dimensional regularisation
(which sends d3p → ddp but retains 3-dimensional mode functions). However, in [47] (see
also [61, 62]), it was pointed out that this log(k) is absent for other regularisations (including
dimensional regularisation with d-dimensional mode functions). We show explicitly in
19Since this arises from a d4x √−g(gµν∇µpi∇νpi)2, there is no explicit η dependence in this interaction.
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appendix C that performing dimensional regularisation with d-dimensional mode functions
introduces an additional log(−ik/H) term, and including this contribution gives a 1-loop
wavefunction coefficient of the form,
ψ1-loopk1k2 =
H2
f4pi
k3
16pi2
[
−γ ipi2 + divergence + local
]
. (5.51)
Note that (5.51) and (5.50) share the same Disc, since Disc [log(k)] = Disc
[
ipi
2
]
= ipi, and
so both are consistent with our cutting rules. This is to be expected, since the cutting rules
use only tree-level data, and therefore are not sensitive to how we have chosen to regulate
the loop divergences.
Physically, we can trace this additional log(−iHk) term in (5.51) back to a logarithmic
divergence near the conformal boundary, limη→0 log (−Hη), which arises in d dimensions.
Such boundary divergences are absent in Minkowski, and so in all of our Minkowski exam-
ples the Disc
[
iψ1-loop
]
implied a log k dependence as in (5.50). But in de Sitter, there can
be divergences both from the loop momenta p → ∞ (which produce log(k)) and near the
boundary η → 0 (which produce log(−ik)). The latter do not affect the Disc, and in fact
such boundary divergences appear already at tree-level for certain values of the mass [7],
but since Disc [log(−ik)] = 0 they are consistent with tree-level unitarity [6, 7].
Perturbative unitarity. From our cutting rules, which fix the value of γ in (5.51), we
can place a lower bound on the size of the one-loop correction to ψk1k2
|ψ1-loopk1k2 | ≥ |Imψ
1-loop
k1k2 | =
|γ|
32pi
H4
f4pi
|ψtreek1k2 | (5.52)
where we have written the imaginary part of (5.51) in terms of ψtreek1k2 = −k3/H2. Pertur-
bative unitarity therefore requires that the γ(cs, c˜3) defined in (5.47) is bounded,
|γ(cs, c˜3)|
f4pi
<
32pi
H4
. (5.53)
Previous bounds on the EFT coefficients using perturbative unitarity have either neglected
numerical coefficients (i.e. treating γ as simply ∼ c˜23) or have worked in a subhorizon
limit (i.e. k  H) where the usual optical theorem for amplitudes can be applied — see
e.g. [51, 52, 63–65] for estimates of the EFT cutoff in that regime. By contrast, (5.53) is
the first precise unitarity bound that genuinely incorporates the effects of the expanding
spacetime, and therefore applies at values of k which are comparable to H.
Phenomenologically, the bound (5.53) has already been overtaken by observational
constraints on c˜3 and cs from the bispectrum. For instance, since cs ≥ 0.021 at 95%
confidence [58], |γ| . 5 × 107 for essentially the whole c˜3 95% confidence interval, while
32pif4pi/H4 ≈ 109. However, the cutting rules have provided more than the bound (5.53):
unitarity has completely fixed Im [ψk1k2 ] at 1-loop, and this has important consequences
for the time dependence of the power spectrum.
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5.4.1 Power spectrum at one loop
The power spectrum Pk1k2 = 〈φˆk1 φˆk2〉 can be computed from the wavefunction coefficients
in the standard way, by performing an average over all field configurations weighted by the
probability density |Ψ[φ]|2,
Pk1k2 =
∫
Dφ |Ψ[φ]|2 φk1φk2 . (5.54)
At weak coupling, we can expand perturbatively in the non-Gaussian wavefunction coeffi-
cients,
Pk1k2 = P treek1k2 + P
1-loop
k1k2 + . . . , (5.55)
where the tree-level result is well-known,
P treek1k2 = −
δ˜3 (k1 + k2)
2 Re
[
ψtree ′k1,−k1
] = |fk1 |2 δ˜3 (k1 + k2) , (5.56)
where we have written ψtree2 in terms of the free theory mode function fk. In every cutting
rule derived above, it is this P treek which should be used — for instance, for a massless
scalar field on de Sitter this corresponds to P treek = H2/2k3.
At 1-loop order, there are corrections from the interactions,
P 1-loopk1k2 = 2|fk1 |4Re
[
ψ1-loopk1k2
]
+
∫
Dφ e−
∫
q
φqφ−q
2|fq |2 φk1φk2
 ∫
q1q2q3q4
2Re [ψq1q2q3q4 ]
4! φq1φq2φq3φq4
+
∫
q1q2q3
q′1q′2q′3
2Re [ψq1q2q3 ]Re
[
ψq′1q′2q′3
]
(3!)2 φq1φq2φq3φq
′
1
φq′2φq′3
 (5.57)
where the
∫ Dφ produces several terms, each of which has a simple diagrammatic inter-
pretation (i.e. all possible pair-wise Wick contractions of the fields). Note that the 1-loop
corrections (and indeed, all L-loop corrections) to the power spectrum are sensitive only
to the real parts of the wavefunction coefficients, Re [ψn]. However, above we have shown
that unitarity is effectively fixing the imaginary parts, Im
[
ψ1-loop2
]
, in terms of tree-level
data. We will now show that Im
[
ψ1-loop2
]
plays an important role in determining the time
dependence of the power spectrum.20 This is not surprising, since unitarity is a constraint
on the time evolution of the system — in particular see appendix A where this aspect of
the cutting rules is made manifest.
20Note that a log k running as in (5.50) would instead produce a one-loop correction to the spectral tilt
of the power spectrum, rather than a one-loop correction to the time dependence. It would be interesting
to investigate whether there is some connection between these two effects.
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Time derivatives of the power spectrum. Defining the bulk power spectrum Pk(η)
via (5.54) using the wavefunction Ψη[φ] evaluated at a finite conformal time, we can expand
near the late-time boundary as,
Pk(η) = Pk(0) + η∂ηPk(0) +
η2
2 ∂
2
ηP (0) +
η3
3! ∂
3
ηP (0) +O
(
η4
)
, (5.58)
where Pk(0) is given in terms of the boundary wavefunction coefficients21 by (5.56)
and (5.57). We will now express the subleading ∂nηPk(0) terms in this expansion in terms
of the wavefunction coefficients, and see that in fact it is ∂3ηP (0) that is fixed by Im [ψ2].
Beginning in the Heisenberg picture, Pk1k2(η) = 〈φˆk1(η)φˆk2(η)〉, the equations of mo-
tion can be used to reduce any ∂nη φˆ to just φˆ and ∂ηφˆ. In particular, near the conformal
boundary the only terms which contribute to the correlator of ∂2η φˆ and any other operator
Oˆ are,
lim
η→0 〈∂
2
η φˆ Oˆ〉 = lim
η→0
〈[2
η
∂ηφˆ+ ∂2i φˆ− Cφ˙(∂φ)2H(∂iφˆ)2
]
Oˆ
〉
, (5.59)
where we have power counted each interaction term using the boundary expansion of the
free field profile,
φ ∼ φ(0)
(
1 + 12k
2η2 +O(η3)
)
+ φ(3)η3
(
1 +O(η2)
)
, (5.60)
for instance the term Cφ˙(∂φ)2Hη∂i
(
φ′∂iφ
) ∼ η2 → 0 near the boundary. The remaining ∂ηφˆ
operators may then be written in terms of the conjugate momentum Πˆ of the Schrödinger
picture, using,
lim
η→0 ∂ηφ = limη→0
[
H2η2 Π + Cφ˙(∂φ)2Hη (∂iφ)
2
]
. (5.61)
This then expresses any limη→0 ∂nηP (η) in terms of equal-time correlators of φˆ and Πˆ.
Following [7], we can simplify such correlators by performing a unitary transformation
of the canonical momentum, Π˜k = Πˆk + Im
[
ψ′k,−k
]
φˆk, which accounts for the free-field
damping and removes any squeezing from the Gaussian state. The 〈φˆΠ˜〉 equal-time corre-
lator can be written simply in terms of the wavefunction coefficients as,
〈φˆk1Π˜k2〉 =
i
2 +
∫
Dφ e−
∫
q
φqφ−q
2|fq |2
− ∫
q1q2q3
Im
[
ψtreeq1q2q3k2
]
3! φq1φq2φq3φk1
+
∫
q′1q′2q′3
Re
[
ψtreeq′1q′2q′3
]
3 φq
′
1
φq′2φq′3
∫
q1q2
Im
[
ψtreeq1q2k2
]
2! φp1φp2φk1
 .
(5.62)
Note that both Im
[
ψtree3
]
and Im
[
ψtree4
]
vanish at the boundary, and so limη→0〈φˆkΠ˜−k〉′→ i2 .
21Note that for a massless scalar field, the bulk wavefunction coefficient is given by,
ψtree = ik
2
H2η
(1− ikη)−1 = ik
2
H2η
− k
3
H2
+ . . .
Extracting the late time limit limη→0 ψtree2 (η) requires a renormalisation of the boundary condition η = 0 (or
equivalently, a Boundary Operator Expansion to replace the bulk φˆ operator with boundary operators)—see
e.g. [7] for details.
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Altogether, this means that we can compute limη→0 ∂nηP (η) by first using (5.59) to
reduce all ∂nη φˆ to φˆ and ∂ηφˆ, and then (5.61) to replace ∂ηφˆ with Πˆ, and finally (5.62) to
simplify the mixed 〈φˆΠˆ〉 correlators. We find that, to quadratic order in the couplings Cφ˙3
and Cφ˙(∂iφ)2 ,
lim
η→0 [Pk(η)] =
H2
2k3 + P
1-loop
k (0) ,
lim
η→0 [∂ηPk(η)] = 0 ,
lim
η→0
[1
2∂
2
ηPk(η)
]
= H
2
2k + k
2P 1-loopk (0)− Cφ˙(∂φ)2H
∫
q1q2
q1 · q2 Btreeq1q2k(0)
lim
η→0
[ 1
3!∂
3
ηPk(η)
]
= −H
4
3k3 Im
[
ψ1-loopk−k
]
, (5.63)
where Bq1q2k(0) = 〈φq1φq2φk〉|η→0 is the boundary value of the bispectrum. So while
the coefficients of η0 and η2 receive divergent loop corrections to their tree-level values
(which require renormalisation), the coefficient of η3 which has been generated by quantum
corrections actually has a fixed finite value, determined by γ.
The fact that it is the third time derivative, ∂3ηP , that is constrained by unitarity seems
to be a result of considering a massless scalar field. In particular, using (5.59) and (5.61),
we have that,
lim
η→0
〈
∂3η φˆk Oˆ
〉
= 2H2 lim
η→0
〈(
Π˜k − Im
[
ψ1-loop ′k,−k
]
φˆk
)
Oˆ
〉
. (5.64)
At tree-level, ∂3ηφ → 2H2Π˜ as η → 0, which comparing with (5.60) is the well-known
result that φ(3) plays the role of the momentum conjugate to φ(0) near the boundary. At
1-loop, Im
[
ψ1-loop2
]
corrects this relation, effectively mixing some φ(0) into the conjugate
momentum. The cutting rules, which follow from unitarity in the bulk, therefore lead to
constraints on the boundary which mix the boundary operator with its shadow (in this
case φ(0) and φ(3)). This is the boundary avatar of bulk unitarity. It would certainly be
interesting to explore this direction further in future.
6 Discussion
In this work, we have derived general Cosmological Cutting Rules for the wavefunction of
the universe on FLRW spacetimes, which enforce the constraints of unitarity to each order
in perturbation theory. Our results are valid for any number of external legs and to any
loop, generalizing previous results obtained in [6, 7] under the banner of the Cosmological
Optical Theorem. Our rules take advantage of a set of algebraic relations that rewrite the
imaginary part of a product of (bulk-to-bulk) propagators in terms of products of factors
with fewer propagators. This reduces the number of nested time integrals that are needed
to compute wavefunction coefficient, which are the main computational obstacle in the
problem. In words, our rules compute a certain discontinuity of loop diagrams in terms
of the discontinuity of diagrams with a lower number of loops. Graphically, our rules
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consists in noticing that the sum over all possible cuts of a given diagram vanishes. This
is very analogous to the flat space cutting rules, but the presence of a boundary term in
cosmological (bulk-to-bulk) propagator makes the details quite a bit different.
We also demonstrate how to use the Cosmological Cutting Rules to derive various one-
loop corrections to the power spectrum from tree-level results. In particular we consider
some simple examples in Minkowski space and then consider the leading cubic and quartic
coupling in the Effective Field Theory of Inflation around quasi de Sitter space. In these
cases, the discontinuity computed by our rules can be interpreted as the coefficient of a
logarithmic corrections to the power spectrum or of its time dependence, depending on the
appropriate physical regularization.
There are a few interesting directions for future research:
• We have illustrated the Cosmological Cutting Rules by applying them to the quadratic
wavefunction coefficient ψ2, but they can be applied more generally to any non-
Gaussian coefficient. Using the general cutting rules presented here to extract infor-
mation about the one-loop correction to ψ4 would be particularly interesting since in
that case unitarity would fix a richer kinematic dependence (in contrast, ψ2 ∼ k3 is
fixed by dilations), and in particular could be combined with a partial wave expan-
sion. This would allow a direct comparison with the existing unitarity bounds on the
subhorizon 2→ 2 scattering amplitude [51].
• The cutting rules were derived here using unitarity in the bulk, and we have shown
that they place constraints on boundary correlators which mix different boundary
operators. Further pursuing this connection with the hypothetical conformal field
theory at the boundary may shed light on what property of the dual holographic
description reproduce unitary dynamics in the bulk.
• In our analysis we restricted to (products of) single discontinuities. But there are
additional relations involving multiple discontinuities that can be derived with similar
techniques. It would be nice to work these out and study the possible relation to the
recently discussed Steinmann relations for the wavefunction [66].
• Our discussion of the implications of unitarity here and in the previous literature [6, 7]
has been perturbative in nature. It would be nice to derive a non-perturbative relation
for the full wavefunction of the universe that in perturbation theory reduces to our
Cosmological Cutting Rules. This could also be tested on recent non-perturbative
results [67]. This would be useful to derive elastic unitarity bounds and perhaps adapt
numerical bootstrap techniques from the amplitude literature (see e.g. [68–70]).
• The analytical structure of cosmological n-point function is relatively unexplored. It
would be nice to understand what kind of functions can arise in general, for example
for a massless scalar or graviton, at tree- and loop level and what their branch points
look like. Our results then relates discontinuities found at different perturbative and
loop orders.
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• Finally, the cutting rules for scattering amplitudes play an important role in deriv-
ing positivity bounds—constraints placed on low-energy EFTs by unitarity (/causal-
ity/locality) of the underlying UV completion [71]. These have recently been devel-
oped in a number of ways for scattering amplitudes on Minkowski spacetime [72–80],
and were recently exported to systems without Lorentz boosts in [81]. In deriving
the Cosmological Cutting Rules, we are now one step closer to realising the same pro-
gram for cosmological correlators. It would be interesting to now combine the cutting
rules presented here with a further study into their analytic structure, developing an
analogous set of UV/IR relations which can be used to translate our measurements
at the end of inflation into properties of the underlying UV physics.
The cutting rules for amplitudes were derived more than half a century ago while we are
only now deriving similar results for cosmological n-point functions. There should be many
other simple and general results that await discovery in this rather unexplored field. We
hope that our progress will bolster the growing interest in this wide open line of research.
Acknowledgments
We would like to thank Harry Goodhew, Mang Hei Gordon Lee, Tanguy Grall, Aaron
Hillman, Austin Joyce, Guilherme L. Pimentel, and David Stefanyszyn for useful discus-
sions. S.M. is supported by an UKRI Stephen Hawking Fellowship (EP/T017481/1) and
partially by STFC consolidated grants ST/P000681/1 and ST/T000694/1. E.P. has been
supported in part by the research program VIDI with Project No. 680-47-535, which is
(partly) financed by the Netherlands Organisation for Scientific Research (NWO).
A Cutting rules from the Schrödinger picture
In the main text, we derived cutting rules from properties of the bulk-to-bulk and bulk-to-
boundary propagators used in the path integral approach. In this appendix we provide an
analogous derivation working directly with the wavefunction in the Schrödinger picture.
This alternative perspective is useful because:
(i) It does not refer to any particular diagram, so would be a natural starting point for
extending these cutting rules to a non-perturbative statement of unitarity,
(ii) It extracts from unitarity (namely H = H†) a number of conserved quantities, which
can be used to connect the boundary observables at η = 0 with properties of the
initial state at η → −∞,
(iii) It demonstrates that the propagator identities which we have made use of the main
text, rather than being an algebraic accident, are actually inevitable properties of
any perturbative solution to the Schrödinger equation.
After outlining the general strategy, we will briefly review the tree-level conserved
quantities that were derived from unitarity in [7]. Then we will show how these quantities
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are corrected at 1-loop. In particular, we will show that in a Bunch-Davies initial state
these conserved quantities reproduce the cutting rule (3.19) for the Disc of iψ2, in terms
of the cubic and quartic wavefunction coefficients, ψ3 and ψ4.
Wavefunction dynamics. The state of the Universe at time η is denoted by |Ψη〉, with
corresponding wavefunction,
Ψη[φ] = exp(iΓη[φ]) , (A.1)
Γη[φ] =
∫
ka
iψk1k2(η)
2!i φk1φk2+
∫
ka
iψk1k2k3(η)
3!i φk1φk2φk3+
∫
ka
ψk1k2k3k4(η)
4!i φk1 . . .φk4+. . . .
In the Schrödinger picture, the ψn(η) coefficients are found by solving their respective
equations of motion, using a set of boundary conditions ψn(η0) inferred from the initial
state |Ψη0〉 (e.g. Bunch-Davies at η0 =∞). The equation of motion for each non-Gaussian
wavefunction coefficient are generated by the functional Schrödinger equation,
−∂ηΓη = 12ad−1
∫
q
(
δΓη
δφq
δΓη
δφ−q
− i δ
2Γη
δφqδφ−q
)
+Hη (A.2)
where we have assumed a canonical kinetic term for φ, a conformally flat background space-
time ds2 = a2(η)
(−dη2 + dx2), and Hη[φ] = 〈φ|Hˆint|Ψη〉 is determined by the interaction
Hamiltonian, Hˆint
[
φˆ, Πˆ
]
, acting on the state. The second derivative, δ
2Γη
δφqδφ−q , formally
diverges (since it requires bringing two operators, Πˆx1 and Πˆx2 , to the same spacetime
point), and is responsible for the loop corrections when (A.2) is solved perturbatively.
Conserved quantities. The aim of the game is to manipulate (A.2) into the form,
∂ηJη = Hη −H†η (A.3)
where H†η[φ] = 〈φ|Hˆ†int|Ψη〉. For unitary evolution, Hη = H†η, this equation becomes a
consversation law, ∂ηJη[φ] = 0. If we similarly expand the functional Jη[φ] as,
Jη[φ] =
∫
ka
βk1k2(η)
2! φk1φk2+
∫
ka
βk1k2k3
3! φk1φk2φk3+
∫
ka
βk1k2k3k4
4! φk1 . . .φk4+. . . (A.4)
then each coefficient βn is constant for any unitary dynamics,
Unitarity ⇒ ∂ηβn = 0 . (A.5)
This is analogous to how the Hamilton-Jacobi equation is used in classical mechanics to
identify constants of motion.
A procedure for deriving the βn was described in [7], where β3 and β4 were constructed
explicitly at tree-level. We will now briefly review that construction, and then extend it to
include loops.
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A.1 Tree-level constants of motion
In the context of (A.2), working at “tree-level” amounts to discarding the −iδ2Γη/δφqδφ−q
term. Taking three or four functional δ/δφ derivatives of (A.2) then produces equations of
motion for ψ3 and ψ4,
∂η
[
iψk1k2k3
∏3
a f
∗
ka
]
∏3
a f
∗
ka
= δ
3Hη
δφk1δφk2δφk3
+ loops (A.6)
∂η
[
iψk1k2k3k4
∏4
a f
∗
ka
]
∏4
a f
∗
ka
= δ
4Hη
δφk1δφk2δφk3δφk4
−
3∑
perm.
1
ad−1
∫
qq′
ψk1k2qψk3k4q′ + loops
where +loops denotes the −iδ2Γη/δφqδφ−q terms we have neglected, and the f∗k mode
functions account for the time-dependence of the field basis, as described in [7]. The sum
in ∂ηψ4 is over inequivalent permutations of the ka momenta, effectively producing s, t
and u exchange contributions.
Working at this order, Hη is given by the interaction Hamiltonian with canonical
momentum replaced by −iψ2φ,
Hη = Hint
[
φk,Πk = −iψ′k,−kφk
]
+ loops , (A.7)
where −iψ′k,−k = ad−1∂ηf∗k/f∗k in terms of the free mode function. The Disc operation
defined in (3.3) has been defined so that Disc [f∗k (η)/f∗k (η0)] = 0, and as a consequence,
Disc
[
δnHη
δφk1δφk2 . . . δφkn
]
= δ
n
δφk1 . . . δφkn
[
Hη −H†η
]
. (A.8)
Equations (A.6) and (A.8) can be immediately combined to give the cubic conserved
quantity. For unitary dynamics, in which Hη−H†η = 0, the Disc of (A.6) can be written as,
∂η β
tree
k1k2k3 = 0 , (A.9)
where,
βtreek1k2k3 = Disc [iψk1k2k3 ]
3∏
a
f∗ka (A.10)
is the conserved quantity at cubic order at tree-level, where we assume that the overall
phase of f∗k has been chosen so that Disc [f∗k ] = 0.
Applying Disc to the ∂ηψ4 equation of motion does not immediately provide β4 because
the Disc [ψ3 × ψ3] is not a total time derivative. This is because the basis of {ψn} coefficients
is not “diagonal” in the following sense: if the initial state has ψ3(η0) 6= 0 and ψ4(η0) = 0,
then at later times ψ4(η) evolves to a non-zero value even in a completely free theory. This
mixing can be removed by defining,
ψ˜k1k2k3k4 = ψk1k2k3k4 +
3∑
perm.
∫
qq′
Pqq′ ψk1k2qψk3k4q′ , (A.11)
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which now has the property that setting ψ˜4(η0) = 0 initially leads to the solution ψ˜4(η) = 0
for all times in the free theory (irrespective of the initial value of ψ3(η0)). The interacting
equation of motion (A.6) for ψ˜4 is then,
∂η
[
iψ˜k1k2k3k4
∏4
a f
∗
ka
]
∏4
a f
∗
ka
= δ
4Hη
δφk1δφk2δφk3δφk4
−i
6∑
perm.
∫
qq′
Pqq′ ψk1k2q
∂η
[
ψk3k4q′f
∗
k3
f∗k4f
∗
q′
]
f∗k3f
∗
k4
f∗q′
+loops
(A.12)
and now the Disc of the ψ3∂ηψ3 exchange contribution is a total time derivative. For a
unitary process, this equation gives,
∂η β
tree
k1k2k3k4 = 0 (A.13)
where the quartic conserved quantity is,
βtreek1k2k3k4 =
4∏
a
f∗ka
{
iDisc [iψk1k2k3k4 ] +
3∑
perm.
∫
qq′
Disc
q
[
iψk1k2q
]
Pqq′ Disc
q′
[
iψq′k3k4
] }
.
(A.14)
Note that for a Bunch-Davies initial state, in which β4(η0) = 0 initially, unitarity requires
that this combination of Disc’s vanishes at any later time, β4(η) = 0. This reproduces
the cutting rule (3.5) derived in the main text from the properties of the bulk-to-bulk
propagator.
This is how the tree-level constants of motion (A.10) and (A.14) were derived in [7].
We are now going to follow the same procedure but retaining the loop corrections from the
δ2Γ/δφqδφ−q term in (A.2). In particular, we will focus on the 1-loop correction to the
quadratic coefficient ψ2, and derive the constant of motion β1-loop2 .
A.2 Loop-level constants of motion
We can expand the quadratic wavefunction coefficient as ψk1k2 = ψtreek1k2+ψ
1-loop
k1k2 +. . ., where
−iψtree ′k−k = ad−1∂ηf∗k/f∗k is the solution to the Schrödinger equation with no δ2Γ/δφqφ−q
term, while ψ1-loopk1k2 is the solution to the equation of motion,
∂η
[
iψ1-loopk1k2 f
∗
k1
f∗k2
]
f∗k1f
∗
k2
= δ
2Hη
δφk1φk2
− 1
ad−1
∫
q
ψtreek1,k2,q,−q , (A.15)
where ψtree4 satisfies (A.6) with no loop terms. It is not difficult to show that the general
solution to this equation can be written in terms of the bulk-to-boundary and bulk-to-
bulk propagators of the main text — for example, for a simple 14!λφ4 interaction in the
Lagrangian, the coefficients,
ψtree ′k1k2k3k4(η0) = +iλ
∫
dηKk1(η, η0)Kk2(η, η0)Kk3(η, η0)Kk4(η, η0)
ψ1-loop ′k−k (η0) = +λ
∫
dηKk(η, η0)Kk(η, η0)
∫
q
Gq(η, η, η0) , (A.16)
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satisfy (A.6) and (A.15), since ∂η0Gq(η1, η2, η0) = −a1−dKk(η1, η0)Kk(η2, η0) (and
δ2Hη/δφ2 = 0 for this theory since there are no quadratic interactions in Hint).
The goal is then take the Disc of (A.15), which effectively removes the interac-
tion term. Note that while there are also loop corrections to the on-shell Hamilto-
nian (A.7), in particular at one-loop order the canonical momentum should be replaced as
Πk = −i
∫
q ψkqφq − i2
∫
q1q2 ψkq1q2φq1φq2 , when Disc
[
ψtree3
]
= 0 is fixed by the tree-level
β3 as in (A.10) this does not change the fact that Disc
[
δ2Hη/δφ2
]
vanishes for unitary
dynamics.
However, the Disc of the ψtree4 term in (A.15) is not a total derivative, so we have not
yet achieved a conservation law. This is because the quartic ψ4 acts as a source for ψ1-loop2
— this means that even in a free theory22 ψ2 will evolve in time for any initial state with
ψ4(η0) 6= 0. We can remove this mixing analogously to the βtree4 example above, by defining
a new wavefunction coefficient,
ψ˜k1k2 = ψk1k2 +
∫
qq′
Pqq′ψk1k2qq′ , (A.17)
in terms of which (A.15) becomes,
∂η
[
iψ˜1-loopk1k2 f
∗
k1
f∗k2
]
f∗k1f
∗
k2
= δ
2Hη
δφk1δφk2
+ i
∫
qq′
Pqq′
∂η
[
ψtreek1k2qq′f
∗
k1
f∗k2f
∗
q f
∗
q′
]
f∗k1f
∗
k2
f∗q f∗q′
, (A.18)
where we have used the following helpful identity [7],
1
f∗q f∗q
= iad−1∂η
(
fq
f∗q
)
. (A.19)
Now when we take the Disc of (A.15), the ψ4 term can be written as a total time derivative
using (A.6), and therefore we find that the Schrödinger equation can be written as,
∂ηβ
1-loop
k1k2 = 0 (A.20)
for unitary dynamics, where this 1-loop constant of motion is given by,
β1-loopk1k2
f∗k1f
∗
k2
= (−i)Disc
[
iψ1-loopk1k2
]
+
∫
qq′
Pqq′(−i)Disc
qq′
[
iψtreek1k2qq′
]
+
∫
q1q′1
q2q′2
Pq1q′1Pq2q′2(−i)Discq1q2
[
iψtreek1q1q2
]
(−i)Disc
q′1q
′
2
[
iψtreek2q′1q′2
]
. (A.21)
Since β2(η0) = 0 in the Bunch-Davies initial state, (A.21) can be used to fix Disc
[
iψ1-loop2
]
in terms of ψtree3 and ψtree4 . This reproduces the combination of both cutting rules (3.15)
and (3.19), since it applies to the full ψ2 (which is a sum over both kinds of diagram
in general).
22One may wonder what we mean by ψ1-loop2 in a free theory. By “free”, we mean that Hint = 0 and there
are no interactions. By “one-loop”, in this appendix we mean next-to-leading order in the small coupling
that suppress non-Gaussianities, i.e. we assume that ψn ∼ gn−2 for some small coupling g.
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Each of the cutting rules given in section 3 can be derived from the Schrödinger
picture in this way: by first writing down the Schrödinger equation of motion for each
ψn, and then using the Disc to remove the contribution from Hint (taking care to remove
any free-theory mixing which arises between the different wavefunction coefficients). It
would be interesting to phrase the general proof of our cutting rules given in section 4.2
in terms of Schrödinger picture dynamics, particularly with regards to formulating a fully
non-perturbative unitarity condition.
B List of propagator identities
In this appendix we list various identities between the real and imaginary parts of the
bulk-to-bulk propagator, Gp(t1, t2), and the bulk-to-boundary propagator, Kp(t).
B.1 Tree-level diagrams
In addition to the relation for a single propagator,
ImGq(t1, t2) = 2PqIm [Kq(t1)] Im [Kq(t2)] (B.1)
the analogous relation for two propagators is,
Im [Gq1(t1, t2)Gq2(t2, t3)] =
2∑
perm.
2Pq1Im [Kq1(t1)] Im [Kq1(t2)Gq2(t2, t3)]
− 4Pq1Pq2Im [Kq1(t1)] Im [Kq1(t2)Kq2(t2)] Im [Kq2(t3)] .
(B.2)
For three-propagators there are two different diagrams: the four-site chain,
Im [Gq1(t1, t2)Gq2(t2, t3)Gq3(t3, t4)] =
+
2∑
perm.
2Pq1Im [Kq1(t1)] Im [Kq1(t2)Gq2(t2, t3)Gq3(t3, t4)]
+ 2Pq2Im [Gq1(t1, t2)Kq2(t2)] Im [Kq2(t2)Gq3(t3, t4)]
−
2∑
perm.
4Pq1Pq2Im [Kq1(t1)] Im [Kq1(t2)Kq2(t2)] Im [Kq2(t3)Gq3(t3, t4)]
− 4Pq1Pq3Im [Kq1(t1)] Im [Kq1(t2)Gq2(t2, t3)Kq3(t3)] Im [Kq3(t4)]
+ 8Pq1Pq2Pq3Im [Kq1(t1)] Im [Kq1(t2)Kq2(t2)] Im [Kq2(t3)Kq3(t3)] Im [Kq3(t4)] (B.3)
and the “flux-capacitor”,
Im [Gq1(t1, t4)Gq2(t2, t4)Gq3(t3, t4)] =
+
3∑
perm.
2Pq1Im [Kq1(t1)] Im [Kq1(t4)Gq2(t2, t4)Gq3(t3, t4)]
−
3∑
perm.
4Pq1Pq2Im [Kq1(t1)] Im [Kq2(t2)] Im [Kq1(t4)Kq2(t4)Gq3(t3, t4)]
+ 8Pq1Pq2Pq3Im [Kq1(t1)] Im [Kq2(t2)] Im [Kq3(t3)] Im [Kq1(t4)Kq2(t4)Kq3(t4)] . (B.4)
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B.2 Single-loop diagrams
For the real part of a loop with one, two, three or four propagators, one can use:
2Re [Gq(t, t)] = 2PqIm [Kq(t)Kq(t)] (B.5)
2Re [Gq1(t1, t2)Gq2(t2, t1)] =
2∑
perm.
2Pq1 Im [Kq1(t2)Gq2(t2, t1)Kq1(t1)]
−4Pq1Pq2Im [Kq1(t1)Kq2(t1)] Im [Kq2(t2)Kq1(t2)] (B.6)
2Re [Gq1(t1, t2)Gq2(t2, t3)Gq3(t3, t1)] (B.7)
=
3∑
perm.
2Pq1Im [Kq1(t2)Gq2(t2, t3)Gq3(t3, t1)Kq1(t1)]
−
3∑
perm.
4Pq1Pq2 Im [Kq1(t2)Kq2(t2)] Im [Kq2(t3)Gq3(t3, t1)Kq1(t1)]
+8Pq1Pq2Pq3 Im [Kq1(t2)Kq2(t2)] Im [Kq2(t3)Kq3(t3)] Im [Kq3(t1)Kq1(t1)]
2Re [Gq1(t1, t2)Gq2(t2, t3)Gq3(t3, t4)Gq4(t4, t1)] (B.8)
=
4∑
perm.
2Pq1Im [Kq1(t1)Kq1(t2)Gq2(t2, t3)Gq3(t3, t4)Gq4(t4, t1)]
−
4∑
perm.
4Pq1Pq2 Im [Kq1(t2)Kq2(t2)] Im [Kq2(t3)Gq3(t3, t4)Gq4(t4, t1)Kq1(t1)]
−
2∑
perm.
4Pq1Pq3Im [Kq1(t2)Gq2(t2, t3)Kq3(t3)] Im [Kq3(t4)Gq4(t4, t1)Kq1(t1)]
+
4∑
perm.
8Pq1Pq2Pq3Im [Kq1(t2)Kq2(t2)] Im [Kq2(t3)Kq3(t3)] [Kq3(t4)Gq4(t4, t1)Kq1(t1)]
−16Pq1Pq2Pq3Pq4Im [Kq1(t2)Kq2(t2)] Im [Kq2(t3)Kq3(t3)] [Kq3(t4)Kq4(t4)] [Kq4(t1)Kq1(t1)]
C Explicit one-loop computation for p˙i3
In this appendix, we describe the computation of the one-loop diagram ψ(a)k1k2 from two p˙i
3
vertices. This corresponds to the integral,
ψ
(a) ′
k,−k =
C2
φ˙3
H2
∫ 0
−∞
dη1
η1
∫ 0
−∞
dη2
η2
∫
p1p2
δ˜3 (p1+p2−k)K ′k(η1)K ′k(η2)G(1,1)p1 (η1,η2)G(1,1)p2 (η1,η2)
(C.1)
where K ′k(η) = ∂ηKk(η) and G
(1,1)
p (η1, η2) = ∂η1∂η2Gp(η1, η2) are the time derivatives of
the propagators (5.33). Since this diagram is symmetric in η1 ↔ η2, we can order η1 > η2
and write this as,
ψ
(a) ′
k,−k = 2H
2C2
φ˙3
∫ 0
−∞
dη1
η1
∫ η1
−∞
dη2
η2
∫
d3p1d3p2
(2pi)3 δ˜
3 (p1+p2−k) E(k,p1,p2,η1,η2) (C.2)
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where E(k, p1, p2, η1, η2) = k4p1p2η21η22ei(k+p1+p2)η2eikη1 sin(p1η1) sin(p2η1). This integral is
divergent, and requires regularisation.
We will use dimensional regularisation, analytically continuing to d = 3 + δ dimen-
sions where the integral is formally finite. Unlike for amplitudes on Minkowski spacetime,
where this procedure is more or less unique, on de Sitter spacetime care must be taken
with specifying precisely how the propagators are to be continued to d-dimensions. This
subtlety arises because, unlike e±ipµxµ on Minkowski (which is a good mode function for
any spacetime dimensions), the de Sitter mode functions in d dimensions are,
fk(η) ∝ (−Hη)d/2H(1)ν (−kη) (C.3)
and its complex conjugate, where H(1)ν is a Hankel function (of the first kind) and ν =√
(d/2)2 − (m/H)2. Using these mode functions leads to d-dependent propagators.
We will first compute (C.2) by analytically continuing to d dimensions with the mode
functions held fixed (i.e. kept at their 3-dimensional value (5.32)), and then compute it in a
scheme which also analytically continues the mode functions. This will result in expressions
of the form (5.50) and (5.51) respectively.
3-dimensional mode functions. The simplest scheme in which to evaluate (C.2) is one
in which the propagators retain their 3-dimensional form (5.33), and only the momentum
integration measures are analytically continued. In this case, the time integrals can be
performed straightforwardly as if in 3 dimensions, giving∫ 0
−∞
dη1
η1
∫ η1
−∞
dη2
η2
E(k, p1, p2, η1, η2)
= k4p1p2 (F (+p1 + p2)− F (+p1 − p2)− F (−p1 + p2) + F (−p1 − p2)) (C.4)
where,
F (q) = −(k + q)
2 + 5(k + q)(k + p1 + p2) + 10(k + p1 + p2)2
(k + p1 + p2)3(2k + p1 + p2 + q)5
. (C.5)
Now we must integrate this over p1 and p2. The d-dimensional integration measure can
be written as,
∫
ddp1ddp2 δd (p1+p2−k) f(k,p1,p2) = Sd−22
∫ ∞
k
dp+
∫ +k
−k
dp−
pd−21 p2
k
f(k,p1,p2) (C.6)
where p± = p1 ± p2, and Sd−2 is the surface area of the (d − 2)-dimensional unit sphere
(i.e. S1 = 2pi). In fact, even before performing these two integrals we can already see the
qualitative form of the solution. If we define pˆ1 = p1/k and pˆ2 = p2/k (and pˆ+ = pˆ1 + pˆ2),
then we have,
ψ
(a) ′
k,−k = H
2C2
φ˙3
S1+δ
(2pi)3+δ k
3+δ I(δ) (C.7)
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where I(δ) is the dimensionless integral,
I(δ) =
∫ ∞
1
dpˆ+
∫ +1
−1
dpˆ−pˆ2+δ1 pˆ
2
2
(
Fˆ (pˆ1 + pˆ2)− Fˆ (pˆ1 − pˆ2)− Fˆ (−pˆ1 + pˆ2) + Fˆ (−pˆ1 − pˆ2)
)
(C.8)
and Fˆ is given by (C.5) with p1,2 replaced by pˆ1,2 and an overall k−6 extracted. Focussing
on only the divergent terms as δ → 0,
I(δ) =
∫ ∞
1
dpˆ+ pˆ
d−3
+
(pˆ+ + 1)6(pˆ+ + 3)4
( 3
16 pˆ
13
+ +
27
8 pˆ
12
+ +
105
4 pˆ
11
+ +
921
8 pˆ
10
+ +
74449
240 pˆ
9
+
)
+ finite
= − 130 δ +O
(
δ0
)
. (C.9)
This produces,
ψ
(a) ′
k,−k = −H2
k3
16pi2
( 2
15C
2
φ˙3
(1
δ
+ log(k)
)
+ local
)
, (C.10)
which coincides with (5.50), and matches the C2
φ˙3
part of γ which was inferred from the
cutting rules in section 5.2 (see equation (5.36)).
While this scheme is computationally very simple, one may worry that the d dimen-
sional quantity which we have computed is in fact not the wavefunction coefficient of any
scalar field theory (it is not a solution of the d-dimensional Schrödinger equation, since
we did not use the d-dimensional propagators). Rather, it corresponds to a purely formal
manipulation of the integral (C.2). We are therefore going to consider a second scheme,
which also analytically continues the mode functions in such a way that the d dimensional
integral that we perform is genuinely computing a wavefunction coefficient of a scalar field
in d spacetime dimensions. We will see that in this second scheme, the log(k) which appears
in (C.10) is absent.
d-dimensional mode functions. Rather than consider a massless scalar field in d
dimensions, for which the mode function (C.3) contains the general Hankel function
H
(1)
d/2(−kη), we will instead consider a scalar field of mass m2 = H2(d2 − 9)/4. This
approaches the massless scalar when d→ 3, and has the simpler mode function,
fk(η) = (−Hη)δ/2 H(1 + ikη)
k
e−ikη√
2k
(C.11)
which differs from the d = 3 mode function for an m = 0 field only by an overall nor-
malisation of (−Hη)δ/2. Similarly, de Sitter invariance requires that the interaction vertex
be analytically continued to Cφ˙3(−Hη)−d+2 (φ′)3. In this scheme, the integral (C.2) is
therefore analytically continued to,
ψ
(a) ′
k,−k = 2H
2C2
φ˙3
∫ 0
−∞
dη1
η1
∫ η1
−∞
dη2
η2
∫
ddp1ddp2
(2pi)3 δ˜
d (p1+p2−k) (H2η1η2)δ/2E(k,p1,p2,η1,η2)
(C.12)
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which differs from the previous scheme by a factor of (H2η1η2)δ/2 in the integrand (note that
the propagators also contain additional O(δ) suppressed terms which we have neglected
since they do not contribute to the divergence).
Now following the same steps as before, we first perform the dη2 and dη1 integrals,
which gives,∫ 0
−∞
dη1
η1
∫ η1
−∞
dη2
η2
(H2η1η2)δ/2E(k,p1,p2,η1,η2)
= H
δk4p1p2
(−i(k+p1+p2))δ (F (+p1+p2)−F (+p1−p2)−F (−p1+p2)+F (−p1−p2)) (C.13)
where F (q) is given in (C.5), and again we have discarded O(δ) terms which will not
contribute to the divergence. Note that this differs from (C.4) by an overall factor of
(−i(k + p1 + p2)/H)−δ.
Finally, we must perform the ddp1ddp2 integrals. We can again use (C.6) to write this
in terms of a dimensionless
∫∞
1 dpˆ+
∫+1
−1 dpˆ− integral,
ψ
(a) ′
k,−k = H
2C2
φ˙3
S1+δ
(2pi)3+δ k
3 (iH)δ I(δ) (C.14)
where we now see that the effect of the additional (H2η1η2)δ/2 factor in the integrand is to
introduce a factor of (iH/k)δ compared with (C.7). Note that, up to O(δ0) finite terms,
this I(δ) coincides with (C.8) and in particular shares the divergence I(δ) = −1/30δ+O(δ).
Therefore we arrive at,
ψ
(a) ′
k,−k = −H2
k3
16pi2
( 2
15C
2
φ˙3
(1
δ
+ log(iH)
)
+ local
)
, (C.15)
in this scheme, which coincides with (5.51), and again successfully reproduces the C2
φ˙3
part
of γ which was inferred from the cutting rules in (5.36). This also agrees with the similar
calculation performed in [47] of the equal-time in-in correlator.
Finally, we note that in this scheme it is crucial to perform the time integrals over dη1
and dη2 before taking the limit δ → 0. In particular, had one expanded (H2η1η2)δ/2 =
1+ δ2 log
(
H2η1η2
)
+ . . . inside the integral, one would have found that there is a logarithmic
boundary divergence, limη→0 log (−Hη), as the late-time boundary is approached. In other
words, in this scheme, the dimensional regularisation is regulating both the p → ∞ UV
divergence from the loop integral and the η → 0 boundary divergence at late times.23
These boundary divergences can arise in the non-Gaussian coefficients even at tree-level,
and were systematically studied in [7] for the cubic wavefunction coefficient (see also [85]).
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
23These boundary divergences should not be confused with the IR (secular) divergences that can appear
in equal-time correlators [41, 82–84]—the wavefunction coefficients never contain these, they arise only
when performing the field average
∫ Dφ|Ψ[φ]|2 over field configurations on a fixed η hypersurface.
– 52 –
J
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