On the EFT of Conformal Symmetry Breaking Kurt HinterbichleraQiuyue LiangbMark Troddenb aCERCA Department of Physics

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On the EFT of Conformal Symmetry Breaking

Kurt Hinterbichler,a,∗Qiuyue Liang,b,†Mark Trodden,b,‡

aCERCA, Department of Physics,

Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106, USA

bCenter for Particle Cosmology, Department of Physics and Astronomy,

University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

Abstract

Conformal symmetry can be spontaneously broken due to the presence of a defect or other

background, which gives a symmetry-breaking vacuum expectation value (VEV) to some scalar

operators. We study the eﬀective ﬁeld theory of ﬂuctuations around these backgrounds, showing

that it organizes as an expansion in powers of the inverse of the VEV, and computing some of the

leading corrections. We focus on the case of space-like defects in a four-dimensional Lorentzian

theory relevant to the pseudo-conformal universe scenario, although the conclusions extend to

other kinds of defects and to the breaking of conformal symmetry to Poincar´e symmetry.

∗kurt.hinterbichler@case.edu

†qyliang@sas.upenn.edu

‡trodden@physics.upenn.edu

arXiv:2210.01139v1 [hep-th] 3 Oct 2022

Contents

1 Introduction 3

2 The EFT of conformal symmetry 4

2.1 Directconstruction..................................... 4

2.2 Comparison to the coset construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Breaking to Poincar´e 7

3.1 2-pointfunction....................................... 8

4 Defect CFT breaking 10

5 Defect CFT 2-point function 12

5.1 Leadingpart ........................................ 12

5.2 Next to leading order corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.3 Two-point reducible diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.4 Cubicloop.......................................... 17

6 Conclusions 23

A Details of the contour choice 24

References 26

1 Introduction

In the real world, no system is inﬁnite, and boundary eﬀects inevitably become important. There-

fore, boundaries and defects in quantum ﬁeld theory are natural subjects of interest in studying

systems with ﬁnite extent or where multiple regions or phases are joined by junctions or localized

impurities. In particular, recent years have seen increased activity in the study of such boundaries

and defects in the context of conformal ﬁeld theory (CFT) (see e.g. [1–3] for overviews.)

A defect in a CFT can be of any dimension. One of the simplest cases is when the defect is

straight. A straight ddimensional defect breaks the Ddimensional conformal group down to the d

dimensional conformal group, plus the group of rotations around the defect. In a Lorentzian CFT,

such a defect can be space-like or time-like, but the more frequently-discussed case is that of a

time-like defect, where the defect represents a spatial boundary or interface in the system.

In contrast, an example where space-like defects are relevant is in some non-inﬂationary early

universe scenarios featuring a pre-big bang phase, in which the universe is nearly ﬂat rather than

accelerating. Some well-known examples of this type are the ekpyrotic scenario [4,5] and genesis-

type models [6,7]. The general class of such models that makes use of a space-like conformal

defect is the pseudo-conformal universe scenario [7–15]. In these scenarios, it is postulated that the

universe before reheating is described by a CFT on a nearly Minkowski spacetime whose conformal

algebra is broken spontaneously by a time-dependent vacuum expectation value (VEV) of some

dimension ∆ scalar primary operator Φ taking the form

hΦi=C∆

(−t)∆,(1.1)

where the dimensionless constant Csignals the strength of the symmetry breaking. The VEV (1.1)

breaks the four-dimensional Lorentzian conformal symmetry down to a three-dimensional Euclidean

conformal symmetry

so(4,2) →so(4,1) .(1.2)

As t→0 from below, the VEV (1.1) diverges and the universe then reheats and transitions to the

standard post-big bang radiation domination phase. The reheating surface at t= 0 is the space-like

defect in the CFT. This CFT scenario can also be a given a ﬁve-dimensional AdS dual description

[16–20]

Our primary interest here will be in studying the eﬀective ﬁeld theory (EFT) that describes

ﬂuctuations around the symmetry breaking vacuum described by (1.1). In [11] an eﬀective ﬁeld

theory for studying such ﬂuctuations was described. Here we will further explore some of the

systematics of this eﬀective theory. In particular, we will see how the EFT expansion organizes

itself as a power series in various powers of 1/C, and we will compute some of the leading corrections

to the 2-point function.

Since our original motivation came from studying the pseudo-conformal universe scenario, we

will specialize in this paper to the case of a space-like co-dimension 1 defect in a four-dimensional

Lorentzian CFT. However, nothing we do depends crucially on this, and everything will generalize

straightforwardly to other dimensions and signatures for both the CFT and the defect.

This EFT can also be used in the simpler case where conformal symmetry is broken to Poincar´e

symmetry via a constant VEV hΦi ∼ f. We will see that the EFT naturally organizes as an

expansion in various powers of 1/f. For example, the two-point function hφ(x)φ(0)iorganizes as

an expansion in various powers of 1/(f x), which is good at long distances (complementary to the

short distance limit which can be probed by the operator product expansion), and that the leading

1-loop quantum correction is universal, independent of the higher derivative operators in the EFT.

Conventions: Dis the spacetime dimension, and we use the mostly plus metric signature. The

curvature conventions are those of [21].

2 The EFT of conformal symmetry

The EFT we seek should describe the ﬂuctuations of ﬁelds around the symmetry breaking VEV

(1.1). This is the EFT that describes the spontaneous breaking of conformal symmetry, and which

was studied many years ago as a prototype for spontaneously broken spacetime symmetries [22,23].

It is equivalent [24–26] to the theory of a co-dimension 1 brane ﬂuctuating in a ﬁxed background

anti-de Sitter space. The same EFT also plays a key role in the proof of the a-theorem [27,28],

and its S-matrix satisﬁes non-trivial soft theorems [29–32].

One well-known way to construct this EFT is from the coset perspective (see e.g. [11,33]).

However, in the following we describe an alternative and more direct method of constructing it for

arbitrary conformal weights, which will prove useful in the rest of the paper, and which starts with

ﬁelds that linearly realize conformal symmetry.

2.1 Direct construction

Our goal is to construct the EFT of a weight ∆ scalar conformal primary ﬁeld Φ. The symmetries

that must be maintained are the usual linearly-realized conformal symmetries,

δΦ = −(xµ∂µ+ ∆)Φ ,(2.1)

δµΦ = −2xµxν∂ν−x2∂µ+ 2xµ∆Φ,(2.2)

which are the scale transformation and special conformal transformations, respectively.

We construct the EFT by writing all conformally invariant terms, order by order in powers of

derivatives. We allow for terms which are non-analytic in the ﬁelds, because we will ultimately be

expanding around a conformally non-invariant VEV1.

Scale invariance is easy to impose; it is equivalent to demanding that each term in the Lagrangian

density has total operator dimension equal to the spacetime dimension D, so that there are no

1This is similar to the philosophy of the “Higgs EFT” as opposed to the “Standard Model EFT” in the context

of electroweak symmetry breaking [34,35].

dimensionful couplings. We will assume throughout that ∆ 6= 0 and D > 2, since other subtleties

arise otherwise.

At zeroth order in derivatives, the only scale invariant term is

L0= ΦD

∆.(2.3)

This term is also invariant under special conformal transformations, so this is our complete zeroth

order Lagrangian.

At second order in derivatives, the only possible scale invariant term, up to total derivatives, is

L2= ΦD−2

∆(∂Φ)2

Φ2.(2.4)

This is also invariant under the special conformal transformations, so this is our 2-derivative La-

grangian.

At fourth order in derivatives, there are three possible scale invariant terms, up to total deriva-

tives:

ΦD−4

∆(Φ)2

Φ2,ΦD−4

∆(∂Φ)2Φ

Φ3,ΦD−4

∆(∂Φ)4

Φ4.(2.5)

However, imposing special conformal invariance, only two linear combinations of these three terms

are invariant. For later convenience we choose these combinations to be

L4= ΦD−4

∆(Φ)2

Φ2−(2∆ −D+ 2)(2∆ −D+ 4)

4∆2

(∂Φ)4

Φ4,

4= ΦD−4

∆(∂Φ)2Φ

Φ3−2∆ −D+ 3

2∆

(∂Φ)4

Φ4.(2.6)

This construction can be continued to all higher orders in derivatives; at each derivative order

there will be some ﬁnite number of independent scale invariant terms up to total derivatives, some

subspace of these will be fully conformal invariant, and a basis of this subspace forms the EFT

Lagrangian at this derivative order.

The full Lagrangian is then the sum of all these terms, with arbitrary coeﬃcients, organized as

a derivative expansion,

L=c0L0+c2L2+c4L4+c0

4L0

4+··· .(2.7)

We will let {c}4≡ {c4, c0

4},{c}6≡ {c6, c0

6, c00

6, . . .}, etc. denote the sets of coeﬃcients of terms at

each derivative order.

2.2 Comparison to the coset construction

The usual approach to constructing this theory is the coset approach, equivalent to the geometric

method as described in [11,33]. Here we will extend this approach to arbitrary ∆ and see that it

is equivalent to the above direct approach.

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OntheEFTofConformalSymmetryBreakingKurtHinterbichler,a;*QiuyueLiang,b;MarkTrodden,b;aCERCA,DepartmentofPhysics,CaseWesternReserveUniversity,10900EuclidAve,Cleveland,OH44106,USAbCenterforParticleCosmology,DepartmentofPhysicsandAstronomy,UniversityofPennsylvania,Philadelphia,Pennsylvania19104,USAAbs...

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On the EFT of Conformal Symmetry Breaking Kurt HinterbichleraQiuyue LiangbMark Troddenb aCERCA Department of Physics

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