Enhanced Soft Limits in de Sitter Space C. Armstrong1 A. Lipstein1 and J. Mei1 1Department of Mathematical Sciences Durham University

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Enhanced Soft Limits in de Sitter Space

C. Armstrong∗1, A. Lipstein†1, and J. Mei‡1

1Department of Mathematical Sciences, Durham University,

Stockton Road, DH1 3LE, Durham, United Kingdom

Abstract

In ﬂat space, the scattering amplitudes of certain scalar eﬀective ﬁeld theories exhibit

enhanced soft limits due to the presence of hidden symmetries. In this paper, we show that

this phenomenon extends to wavefunction coeﬃcients in de Sitter space. Using a represen-

tation in terms of boundary conformal generators acting on contact diagrams, we ﬁnd that

imposing enhanced soft limits ﬁxes the masses and four-point couplings (including curva-

ture corrections) in agreement with Lagrangians recently derived from hidden symmetries.

Higher-point couplings can then be ﬁxed using a bootstrap procedure which we illustrate at

six points. We also discuss implications for the double copy in de Sitter space.

∗connor.armstrong@durham.ac.uk

†arthur.lipstein@durham.ac.uk

‡jiajie.mei@durham.ac.uk

arXiv:2210.02285v2 [hep-th] 14 Dec 2022

Contents

1 Introduction 2

2 Review 4

2.1 Symmetries and Soft Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 deSitterLagrangians.................................. 6

2.3 Wavefunction Coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Four-point soft limits 9

3.1 NLSM .......................................... 10

3.2 DBI ........................................... 11

3.3 sGal ........................................... 13

3.4 DoubleCopy ...................................... 15

4 Higher Points 17

4.1 NLSM .......................................... 18

4.2 DBI ........................................... 19

5 Conclusion 21

A 4-point sGal Soft Limit 23

B 6-point DBI Soft Limit 25

C Matching 6-point Wavefunctions 27

1 Introduction

There is a deep relation between soft limits of scatteing aplitudes and hidden symmetries. For

example, the soft theorems of graviton amplitudes [1–3] encode extended BMS symmetry [4,

5], while soft limits of pion amplitudes encode spontaneously broken chiral symmetry of QCD

[6]. Pions are the Goldstone bosons associated with spontaneous symmetry breaking and are

described by a low-energy eﬀective action known as the non-linear sigma model (NLSM) [7–9].

Of particular interest for this paper is a property of NLSM amplitudes known as the Adler

zero [10], which is an example of an enhanced soft limit. A scattering amplitude is said to

exhibit an enhanced soft limit when it scales like O(pσ), where pis the soft momentum and

σis an integer greater than the expectation based on counting the number of derivatives per

ﬁeld in the Lagrangian. For the NLSM, σ= 1. More generally, σcan be no higher than three

and the cases σ= 2,3correspond to the Dirac-Born-Infeld (DBI) and special Galileon theories,

respectively [11, 12]. Enhanced soft limits arise from cancellations among Feynman diagrams of

diﬀerent topology and are a consequence of symmetries [11, 13]. In the NLSM, this is just an

ordinary shift symmetry but in the other two cases the symmetries are higher shift symmetries

which are nontrivially realised from the point of view of the Lagrangian and are often referred

to as hidden symmetries.

Soft limits also play an important role in cosmology. For example in the context of inﬂa-

tion, where the early universe is approximately described by de Sitter space (dS), they provide

constraints relating higher-point correlators to conformal transformations of lower-point correla-

tors [14–16], and certain inﬂationary 3-point functions can be deduced from soft limits of 4-point

de Sitter correlators [17–21]. Lagrangians for DBI and sGal theories were also recently deduced

from higher shift symmetries in dS [22]. These Lagrangians have nontrivial masses and curva-

ture corrections away from the ﬂat space limit. As we will see in this paper, the NLSM can be

trivially uplifted to dS space since curvature corrections would break the shift symmetry. It is

therefore natural to ask if the wavefunction coeﬃcients of these theories (which can be computed

from Witten diagrams ending on the future boundary of dS [14, 23–26]) exhibit enhanced soft

limits analogous to their scattering amplitudes in the ﬂat space limit. The goal of this work is

to provide evidence that this is indeed the case. Wavefunction coeﬃcients for the NLSM, DBI,

and sGal theories were previously studied in ﬂat space, but they do not exhibit enhanced soft

limits in this background [27].

To study soft limits of wavefunction coeﬃcients, it is natural to work in dS momentum space

[28–30], which is also the standard language used for cosmology (see [24, 31–46] for some recent

developments). Another technique we will employ is to express the wavefunction coeﬃcients in

terms of boundary conformal generators acting on contact diagrams [47–54]. Soft limits can then

computed by Taylor expanding bulk-to-boundary propagators in the contact diagram, acting on

them with boundary conformal generators, and using the equations of motion to remove terms

which are not linearly independent. Starting with a general eﬀective action with unﬁxed masses

and couplings (including curvature corrections), we then ﬁnd that imposing enhanced soft limits

of the tree-level 4-point wavefunction coeﬃcients ﬁxes all the masses and 4-point couplings for

the DBI and sGal theories in agreement with the Lagrangians constructed in [22]. For the NLSM,

we ﬁnd that enhanced soft limits forbid mass terms or curvature corrections, so the Lagrangian

can be trivially lifted from ﬂat space. These results in turn allow us to ﬁx all the parameters

of the generalised double copy prescription proposed in [50], which relates the 4-point tree-level

wavefunction coeﬃcient of the NLSM model to those of the DBI and sGal theories 1. Above four

points, there must be non-trivial cancellations between contact and exchange Witten diagrams in

order to have enhanced soft limits. Since lower-point couplings feed into the exchange diagrams,

in principle this allows us to ﬁx all higher-point couplings using a bootstrap procedure, which

we demonstrate for the NLSM and DBI theory at six points. The method can also be applied to

the sGal theory above four points, but the Witten diagrams become very numerous so we save

that for future work.

This paper is organised as follows. In section 2, we review enhanced soft limits and their

underlying symmetries in the context of scattering amplitudes, the Lagrangians for the NLSM,

DBI, and sGal theories in dS, and the method for computing wavefunction coeﬃcients in terms

of boundary diﬀerential operators acting on contact diagrams. In section 3, we use enhanced soft

limits to ﬁx the masses and 4-point couplings of the NLSM, DBI, and sGal theories in dS, and

comment on the the double copy of 4-point wavefunction coeﬃcients. In section 4, we describe

the extension of this procedure to higher points and use it to ﬁx all 6-point couplings of the

NLSM and DBI theory. We then present our conclusions and future directions in section 5.

There are also three Appendices containing details about our four and six-point calculations.

2 Review

In this section we will brieﬂy review enhanced soft limits and their relation to shift symmetries

in ﬂat space, mainly following [11]. We then review the Lagrangians for the NLSM, DBI, and

sGal theories in dS and explain how to compute cosmological wavefunction coeﬃcients.

2.1 Symmetries and Soft Limits

Let us ﬁrst consider a scalar ﬁeld theory with the following global shift symmetry:

δφ =θ. (2.1)

The ﬁeld φis a Goldstone boson and can be produced from the vacuum by acting with the

Noether current associated with the shift symmetry:

hφ(p)|Jµ(x)|0i=ipµF eip·x,(2.2)

1The double copy was ﬁrst proposed in the context of scattering amplitudes, relating graviton amplitudes

to the square of gluon amplitudes [55, 56]. For recent work on the double copy for (A)dS correlators see for

example [53, 57–68].

where the right-hand-side is ﬁxed by Lorentz invariance up to an overall constant F. Inserting

the current between incoming and outgoing states then gives

hout|Jµ(0) |ini=−pµ

p2Fhout +φ(p)|ini+Rµ(p),(2.3)

where pµis the diﬀerence between the momenta of the in and out states. The ﬁrst term on the

right hand side contains a pole for the emission of a Goldstone boson and we assume that p·R

vanishes as pµ→0(which requires the absence of cubic vertices involving the Goldstone boson).

Multiplying by pµand noting that the current is conserved up to contact terms which do not

contribute to the scattering amplitude after LSZ reduction, we ﬁnd that

hout +φ(p)|ini=1

Fp·R. (2.4)

From this we immediately see that the amplitude for φproduction vanishes in the soft limit:

lim

p→0hout +φ(p)|ini=O(p).(2.5)

This is the famous Adler zero [10].

Now let’s consider a scalar theory with a higher shift symmetry:

δφ =θµ1...µkxµ1...xµk+..., (2.6)

where θis a constant and the ellipsis denote ﬁeld-dependent terms that we will not need to

consider. This can be thought of as a special case of the shift in (2.1) after promoting θto a

function of position, which is the standard way to compute the Noether current. As a result, one

ﬁnds that the Noether current associated with (2.6) is roughly speaking obtained by dressing the

Noether current associated with (2.1) with xµ1...xµk. Inserting the new current between in and

out states and repeating similar steps to the argument above, one then ﬁnds that the soft limit

of hout +φ(p)|inivanishes after being acted on by kmomentum derivatives which arise from

Fourier transforming xµ1...xµkto momentum space, implying a higher-order Adler zero:

lim

p→0hout +φ(p)|ini=Opk+1.(2.7)

The NLSM, DBI, and sGal theories correspond to k= 0,1,2, respectively. This behaviour

arises from nontrivial cancellations among Feynman diagrams and is therefore referred to as an

enhanced soft limit.

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EnhancedSoftLimitsindeSitterSpaceC.Armstrong*1,A.Lipstein1,andJ.Mei11DepartmentofMathematicalSciences,DurhamUniversity,StocktonRoad,DH13LE,Durham,UnitedKingdomAbstractInatspace,thescatteringamplitudesofcertainscalareectiveeldtheoriesexhibitenhancedsoftlimitsduetothepresenceofhiddensymmetries.In...

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Enhanced Soft Limits in de Sitter Space C. Armstrong1 A. Lipstein1 and J. Mei1 1Department of Mathematical Sciences Durham University

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