Fifth forces and frame invariance Jamie Bamber1 1Astrophysics University of Oxford Denys Wilkinson Building

2025-04-27 0 0 1.62MB 12 页 10玖币

侵权投诉

Fifth forces and frame invariance

Jamie Bamber1, ∗

1Astrophysics, University of Oxford, Denys Wilkinson Building,

Keble Road, Oxford OX1 3RH, United Kingdom

(Dated: Received October 13, 2022; published – 00, 0000)

I discuss how one can apply the covariant formalism developed by Vilkovisky and DeWitt to obtain

frame invariant ﬁfth force calculations for scalar-tensor theories. Fifth forces are severely constrained

by astrophysical measurements. It was shown previously that for scale-invariant Higgs-dilaton grav-

ity, in a particular choice of Jordan frame, the dilaton ﬁfth force is dramatically suppressed, evading

the observational constraints. Using a geometric approach I extend this result to all frames, and

show that the usual dichotomy of “Jordan frame” versus “Einstein frame” is better understood as

a continuum of frames: submanifold slices of a more general ﬁeld space.

I. INTRODUCTION

Since Einstein formulated his theory of General Rel-

ativity over a century ago [1] there has been much the-

oretical interest in the possibility that it is merely an

approximation to a more general theory of gravity. One

of the most popular classes of theories of modiﬁed grav-

ity are the so-called “scalar-tensor” theories [2], where

the Einstein-Hilbert action 1

S=Zd4x√−gM2

2R+Lm,(1)

is modiﬁed by the addition of one or more scalar ﬁelds.

Lmis the matter part of the Lagrangian2. Instead of a

ﬁxed Planck mass MPl (or alternatively a ﬁxed Newton’s

constant G) we introduce a non-trivial coupling to R,

giving an action of the form

S=Zd4x√−g"F(ϕ)R−1

2∂µϕ·∂µϕ−W(ϕ) + Lm#,

(2)

where the eﬀective Planck mass is now a function of the

scalar ﬁeld(s) ϕ={φi}. The ﬁrst and arguably simplest

theory of this type is that of Brans-Dicke from 1961 [3]

where F(φ) = −α

12 φ2, W = 0 for a single scalar ﬁeld φ

and constant α. A modern formulation which encom-

passes all possible scalar-tensor theories with second or-

der equations of motion was given by Horndeski [2, 4].

One feature of these theories is that they can be ex-

pressed in diﬀerent guises or “frames” via ﬁeld redef-

initions. Equation (2) describes a “Jordan” frame if

F(ϕ) depends on ϕ. With a suitable Weyl transforma-

tion gµν →Ω2(ϕ)˜gµν we can obtain a new action S=

∗james.bamber@physics.ox.ac.uk

1note that throughout we assume a mostly plus signature

(−,+,+,+).

2There is sometimes ambiguity as to whether the Lagrangian is

deﬁned with or without the √−gterm. I will be using curly

Lto refer to Lagrangian including the √−gmetric factor, and

upright Lwhen not including the √−g.

Rd4x√−˜ghM2

2˜

R+··· +Lm(Ω(ϕ),˜gµν ,matter)i, where

the gravity sector is now as in GR, but the matter sector

picks up additional couplings to ϕ. This is termed the

“Einstein frame”.

Despite the theoretical attractions (Paul Dirac argued

for a dynamical Gon the basis of his large number hy-

pothesis) generic scalar-tensor theories are severely con-

strained by solar system and lab observations. This is

because the introduction of an additional ﬁeld with a

non-trivial coupling to gravity or matter can in general

mediate long-range “ﬁfth forces” [3, 5, 6]3. The exchange

of a new particle of mass mcoupling to matter gives a

Yukawa [7] potential

Vﬁfth(r) = −2M1M2

4πr e−mr,(3)

for coupling and masses M1, M2. For small enough m

(m= 0 for Brans-Dicke) this can be probed via solar

system tests, which put extremely tight bounds on [8–

10]. In other words if a theory predicts a signiﬁcant long-

range ﬁfth force, like standard Brans-Dicke, it is probably

ruled out.

One particularly interesting type of scalar-tensor the-

ory is one that is scale-invariant (including “Higgs-

dilaton” theories where one of the scalar ﬁelds is a non-

minimally coupled SM Higgs boson) [11–32], where the

action, including the matter sector, has a global Weyl

symmetry [33] such that there is no a-priori lengthscale.

Instead the symmetry is broken dynamically as the scalar

ﬁeld(s) tend towards ﬁxed equilibrium values under the

inﬂuence of an expanding cosmology. This has been pro-

posed as one element of a solution to the so-called hi-

erarchy problem [16, 30, 34], and the phenomenological

implications of such a theory have also generated sub-

stantial interest [35–42]. There is a massless Goldstone

boson associated with the spontaneously broken symme-

try, termed the “dilaton”, σ, and as such we might be

worried about ﬁfth-force constraints. However, Ferreira,

3So-called because they act in addition to the standard four fun-

damental forces of nature.

arXiv:2210.06396v1 [gr-qc] 12 Oct 2022

Hill & Ross (2016) [43] show that, in a particular choice of

Jordan frame, the dilaton completely decouples from the

matter sector, and therefore contributes no ﬁfth force.

This leads to the question: does this result apply

in other choices of frame? Indeed, to what extent are

generic scalar-tensor theories of this type really physi-

cally equivalent in diﬀerent frames? While on a classical

level one should not expect a redeﬁnition of variables to

change the physics or physical results, once you include

quantum corrections this becomes no longer obvious (this

is sometimes called the “cosmological frame problem”

[44]). Copeland et al. [45] and Burrage et al. [46] ex-

plicitly calculated the ﬁfth forces for a three-scalar-ﬁeld

toy model, which becomes a Higgs-dilaton theory for a

certain choice of parameters, in ﬁrst the Einstein frame

[45] and a Jordan frame [46], and showed that at lowest

perturbative order the results are the same. There has

also been extensive work examining the general question

of frame (in)equivalence from numerous points of view,

mostly focused on cosmological applications [47–69].

In particular, Falls & Herrero-Valea [69–71] and Finn

et al. [72, 73] developed a formalism to characterise ex-

actly how the quantum eﬀective action must transform

non-trivially between frames. Finn et al. [72, 73] adopts

the covariant approach [74–76], pioneered by Vilkovisky

and DeWitt [77–79], whereby frame transformations are

described in terms of a coordinate changes on a ﬁeld-

space manifold, and constructs a fully covariant quantum

eﬀective action, extending the Vilkovisky-DeWitt unique

eﬀective action to theories with fermion ﬁelds.

In this paper I show how this formalism, and the co-

variant geometric approach, can be applied to the prob-

lem of computing ﬁfth forces, and to scale invariant

scalar-tensor theories in particular. I extend the geomet-

ric approach to show how choices of frame can be charac-

terised in a geometric manner: as choices of submanifold

in a higher dimensional general ﬁeld space. Frame in-

variance becomes manifest, and we see how the choice

of frame is better thought of not as a dichotomy be-

tween “Jordan” and “Einstein”, but as a continuum one

can smoothly traverse. We also see how scale-invariant

scalar-tensor gravity evades ﬁfth force constraints in all

possible frames.

The structure of this paper is as follows. Section II lays

out the background theory. Section III describes the new

geometric approach to frame ﬁxing, and in section IV I

apply it to calculations of ﬁfth forces. I focus on the scale

invariant theory in section V, and brieﬂy discuss one-

loop corrections from the choice of physical spacetime in

section VI. Finally I conclude with a discussion of my

results and future directions.

II. BACKGROUND

A. The dilaton and scale invariant gravity

Under the Weyl transformation gµν = Ω2˜gµν the Jor-

dan frame action (2) for some integer number of scalar

ﬁelds ϕ={φi}becomes

S=Zd4xp−˜gF(ϕ)Ω2˜

R−6( ˜

∇ln Ω)2−6˜

ln Ω

−Ω21

∂µφi∂µφi−Ω4W(ϕ)+Ω4Lm,

(4)

where ( ˜

∇v)2:= ( ˜

∇µv)( ˜

∇µv). Let

Ω = exp(σ),˜

F(σ, ˜

ϕ) = Ω2F(ϕ),˜

φi= Ωφi,

K=1

φ2+ 6 ˜

F , ˜

W(σ, ˜

ϕ)=Ω4W(ϕ),˜

Lm= Ω4Lm,

(5)

Then we obtain

S=Zd4xp−˜gh˜

F(σ, ˜

ϕ)˜

R−6( ˜

∇σ)2−6˜

σ

−1

φ2

i(˜

∇σ)2+ ( ˜

∇µσ)X

φi˜

∇µ˜

φi

−1

∂µ˜

φi∂µ˜

φi−˜

W(σ, ˜

ϕ) + ˜

Lmi.

(6)

Integrating by parts gives

S=Zd4xp−˜gh˜

F(σ, ˜

ϕ)˜

R−˜

K(σ, ˜

ϕ)( ˜

∇σ)2

+˜

∇µσ˜

∇µ˜

K(σ, ˜

ϕ)−1

∂µ˜

φi∂µ˜

φi−˜

W(σ, ˜

ϕ) + ˜

Lmi.

(7)

The Euler-Lagrange equation for the dilaton σgives

˜

K−2( ˜

K−∂σ˜

K)˜

σ−2˜

∇µ(˜

K−∂σ˜

K)˜

∇µσ=

=−∂σ˜

W−˜

R∂σ˜

F , (8)

where ∂σ:= ∂

∂σ . We can see from the transformation

rules (5) that if we choose F(ϕ) to be quadratic in φi,

the potential W(ϕ) to be quartic in φi, and the Lmto

also rescale appropriately, then the action becomes scale

invariant and ∂σ˜

F=∂σ˜

W= 0. The dilaton is then

massless, and appears in the action only via its deriva-

tives.

S=Zd4xp−˜gh˜

F(˜

ϕ)˜

R−˜

K(˜

ϕ)( ˜

∇σ)2

+˜

∇µσ˜

∇µ˜

K(˜

ϕ)−1

∂µ˜

φi∂µ˜

φi−˜

W(˜

ϕ) + ˜

Lmi.

(9)

If we then choose Ω such that ˜

K=const. we obtain the

particular Jordan frame described in [43] and we see that

the dilaton completely decouples from the other scalar

ﬁelds and the other matter terms in ˜

Lm. The equation of

motion for the dilaton reduces to a wave equation ˜

σ=

0, and it can be set to zero. As a result there are no ﬁfth

forces from the dilaton4.

B. The covariant formalism

Start by considering the path integral

Z[J] = Z[DNΦ]e−S[ϕ]−JaΦa,(10)

where [DNΦ] is an appropriate measure over the func-

tion space for ﬁelds Φ={Φi},Jais a source term. I use

i, j . . . indices to denote ﬁeld species and a, b . . . to de-

note DeWitt indices spanning both ﬁeld species and posi-

tion or momentum [70]. A frame transformation is a ﬁeld

reparameterisation Φi→˜

Φi(Φ). In the covariant formal-

ism we describe this as a transformation of coordinates

on a “conﬁguration space” or “ﬁeld space” manifold [73].

This has an associated line element ds2=CabdΦadΦb

where a, b. We would like our path integral, action and

measure to be frame/reparameterisation invariant, which

leads us to deﬁne

[DNΦ] = V−1

gaugepdet(Cab)Πa

dΦa

√2π,(11)

where Vgauge =RΠadξa

√2πpdet(σab(Φ)) accounts for the

volume of the gauge group, where dξaare the generators

of the Lie algebra and σab is another metric. For the

moment I assume all the ﬁelds are bosonic, however this

formalism has also been extended to include fermionic

ﬁelds [73] as I discuss later. To ensure diﬀeomorphism

invariance of the free action, the preferred ﬁeld space

metric for four dimensions is [72, 73]

Cab =¯gµν

δ2S

δ(∂µΦa)δ(∂µΦb)=Cij ¯

δ(4)(xa−xb),(12)

where the delta function enforces locality. We assume

σab =σµν ¯

δ(4)(xa−xb) is also ultra-local [70]. The ¯gµν is

the physical, or preferred, spacetime metric which satis-

ﬁes dimensionless line element d¯s2= ¯gµν dxµdxν.Deﬁn-

ing ¯gµν is important to overcome the ambiguity between

the physical space time metric and the gravity quantum

ﬁeld gµν [70, 72, 73]. The two are related by

¯gµν =l−2(Φ)gµν ,(13)

=e−2σphys gµν ,(14)

=e2(σ−σphys)˜gµν ,(15)

4There is still a coupling between gravity and the dilaton via its

contribution to the stress energy tensor Tµν , and thus sourcing

curvature according to standard General Relativity, however this

contribution also vanishes for σ= 0.

where l(Φ) is an eﬀective Planck length [73], and the

functional derivatives are deﬁned using the barred metric,

and ¯

δ(4)(x) is deﬁned such that Rd4x√¯g¯

δ(4)(x) = 1. For

an action with kinetic term

S=Zd4x−Nij ˜gµν ∂µΦi∂νΦj+. . . ,(16)

(summation implied) the associated ﬁeld space metric is

Cij =e2(σ−σphys)Nij .(17)

In theories without gravity we can take σphys =σand

canonically normalise the kinetic term so that Nij =

const., allowing us to neglect the ﬁeld space metric en-

tirely as it only contributes a overall constant to the path

integral. However, in theories with gravity the choice of

σphys is important, and Cij can have non-trivial depen-

dence on the ﬁelds Φi. In order to obtain perturbative

scattering amplitudes we expand about a ﬂat Minkowski

background gµν ≈ηµν +hµν . With a trivial ﬁeld space

metric we can use the background ﬁeld approach to ob-

tain Feynman rules with vertex coeﬃcients and propaga-

tors given by

λab...c =i ∂(a∂b. . . ∂c)S,(18)

∆ab =i h∂a∂bSi−1,(19)

where a, b... are once again DeWitt indices in either posi-

tion or momentum space, the factors of i come from the

Wick rotation and h. . . idenotes (. . . )|Φ=Φ0setting the

ﬁelds to their background or equilibrium values Φ0. The

(a, b...c) denotes symmeterisation over the indices. To

incorporate the ﬁeld space determinant we can take ei-

ther of two approaches: either work out the determinant

contribution to the eﬀective Lagrangian via

pdet(Cab) = exp 1

2Tr (ln (Cab)),(20)

= exp 1

2Zd4x√¯gTr (ln (Cij (x))),(21)

then expand in powers of the coupling constants [71]. Al-

ternatively one can modify the Feynman rules by promot-

ing partial derivatives to covarient ﬁeld space derivatives

[72, 73]

λab...c →i∇(a∇b. . . ∇c)S,(22)

∆ab →ih∇a∇bSi−1.(23)

As Sis a ﬁeld space scalar, the n-vertex λab...c is a 0

n

rank ﬁeld space tensor, and the propagator ∆ab is 2

0

tensor. For Feynman diagrams with external legs we also

need to deﬁne the external factor

Xa=∂Φa

∂χ ,(24)

where χis the physical external ﬁeld connected to that

leg, a ﬁeld space scalar. This makes Xaa ﬁeld space

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

FifthforcesandframeinvarianceJamieBamber1,1Astrophysics,UniversityofOxford,DenysWilkinsonBuilding,KebleRoad,OxfordOX13RH,UnitedKingdom(Dated:ReceivedOctober13,2022;published{00,0000)IdiscusshowonecanapplythecovariantformalismdevelopedbyVilkoviskyandDeWitttoobtainframeinvariantfthforcecalculationsf...

展开>> 收起<<

Fifth forces and frame invariance Jamie Bamber1 1Astrophysics University of Oxford Denys Wilkinson Building.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Fifth forces and frame invariance Jamie Bamber1 1Astrophysics University of Oxford Denys Wilkinson Building

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: