Transverse(Diff) theories of gravity

Madrid Investiga iones Cientí as

Fa ultad de Cien ias Instituto de Físi a Teóri a

Departamento de Físi a Teóri a IFT UAM-CSIC

Transverse(Di) Theories of Gravity

Juan José López Villarejo

Proye to dirigidopor

Enrique Álvarez Vázquez

Madrid, O tubre 2010.

from the realm of reality.

Bus a lo imposible,

desde el mundo de lo real.

A Jos y Belén, y muy espe ialmentea Enrique, por su empeño y esfuerzo guiándome,

desde su experien ia y en ada su esiva etapa, en la dire ión más ade uada para mi

progreso omoinvestigador. Todoelloa pesar de habertopado on un difí ilperl de

do ente y "lósofo".

AJesúsyMaríaLuisa,pormantenerlamoral"delatropa"siemprealta. AMariola

y Luigi.

Y a todas las personas que, a nivel profesional o personal, me han ayudado en

alguna forma a al anzar este resultado. Cito algunos: Miguel Arana, Pepe Barbón,

CarlosBar eló, DiegoBlas,AntónF. Faedo,LuisGaray,CarloMa, José M a

Martín,

Guillermo Mena, David Morton, Josep Pons, Roberto Rubio, Javi Sabio, M a

Angeles

Vozmediano, DanielZenhäusern. A todos, mu has gra ias!

We work in natural units, for whi h

c = ~ = 1

^{. For the main part, we adopt the}

timelike Landau-Lifshitz onventions. These onventions may vary at some points,

where itis expli itlywarned.

The signature of the metri is

(+, −, −, −)

^.

The Riemanntensor is

R ^µ ναβ ≡ ∂ ^α Γ ^µ _νβ − ∂ ^β Γ ^µ _να + Γ ^µ _σα Γ ^σ _νβ − Γ ^µ σβ Γ ^σ _να

and the Ri itensor

R µν ≡ R ^{λ µλν} .

1 Introdu tion and outline 1

1.1 Introdu tion . . . 1

1.2 Outline. . . 8

2 Ultraviolet behavior of TDi gravity 11 2.1 Introdu tion . . . 11

2.2 The one-looptransverse ee tive a tion . . . 12

2.3 A more generaltransverse a tion . . . 17

2.4 Con lusions . . . 20

2.5 Appendix: te hni al notes . . . 21

2.5.1 Some details of the omputations . . . 21

3 TDi gravity vs. observations 31 3.1 Introdu tion . . . 31

3.2 The Prin ipleof Equivalen e . . . 34

3.3 Masses in transverse theories. . . 36

3.4 Commentson the Newtonian limit . . . 43

3.5 Con lusions . . . 44

3.6 Appendix: te hni al notes . . . 47

3.6.1 Transverse energy-momentumtensors and their onservationlaws. 47 3.6.2 Conservation of the a tive gravitationalmass. . . 49

4.1 Introdu tion . . . 53

4.2 TDi Gravity vs. Unimodular theories . . . 56

4.3 Corresponden e between TDi gravity and a generals alar-tensortheory. 57 4.4 Energy-Momentum Tensors . . . 63

4.5 Flat Limit . . . 64

4.5.1 Global at limit(weak eld) . . . 64

4.5.2 Lo alat limit(freefall).. . . 65

4.6 Con lusions . . . 66

4.7 Appendix: te hni al notes . . . 66

4.7.1 A tive symmetry groupvs. passive ovarian e group. . . 66

4.7.2 Unimodular gravity vs General Relativity . . . 67

4.7.3 Gauge transformationbetween

φ

^and

√ g − 1

^{. . . . . . . . . . . 68}

5 Con lusions 69

Introdu tion and outline

1.1 Introdu tion

Symmetryprin ipleshaveagreatrelevan einmodernphysi s. Thesu essfulStandard

Modelofparti lephysi sisbaseduponagaugesymmetry group

U(1)×SU(2)×SU(3)

^,

whi h is spontaneously broken to a

U(1) × SU(3)

^{. The pre ise me hanism for the}

breakingofthesymmetrywillbehopefullyunraveledbythepresentexperimentsatthe

Large HadronCollider, LHC,inGeneva (Switzerland). TheStandard Model orre tly

a ounts for ele tromagneti , weak and strong intera tions, although it is known not

to be validfor arbitrarilyhigh energies beyond the (redu ed) Plan k mass s ale 1

M p ≡

r ~ c

8πG ≈ 2.43 × 10 ¹⁸ GeV · c ⁻² .

It is thus to be onsidered as a (quantum) ee tive eld theory. On its part, the

gravitational intera tion has not been traditionally integrated into this pi ture due

to its perturbative non-renormalizability; however, it an again be understood as an

ee tive eld theory that leads to sensible predi tions at energies mu h below

M p

[21,36℄.

The standard and best-motivated theory of gravitation atthe lassi al level  as

opposed to quantum  is Einstein's General Relativity (GR in what follows). Its

equations relate the energy-momentum tensor of matter

T µν

^{with a measure of the}

1

Wehavekept

~ , c

^{here. Sour e: NIST,}http://www.nist.gov/pml/data/physi al onst. fm

urvature of spa etime

G µν

^{through Newton's onstant}

G

^{ and with the possible}

o urren e of a osmologi al onstant

Λ

^:

G µν + Λg µν = κT µν with κ ≡ 8πG ≡ M p ⁻² ,

^(1.1.1)

where

G µν ≡ R ^µν − ¹ ₂ Rg µν

^{, with}

R

^{the Ri i s alar and}

R µν

^{the Ri i tensor. These}

equations are onstrained by the so- alled Bian hiidentities:

∇ ^µ G ^µν = ∇ ^µ T ^µν = 0.

^(1.1.2)

GR has been tested to an in redible pre ision by many dierent experiments (see

e.g. [83℄ and [39℄). It does present however some short omings at the osmologi al

level,wherethe originof the vast majorityof energy density intheUniverse isnot yet

understood(roughly,5%ofordinarymattervs. 70%darkenergy+25%darkmatter).

There is strong eviden e that 'dark matter' an not be a ounted for by means of

a modied theory of gravity, as the Bullet Cluster [23℄ and the more re ent MACS

J0025.4-1222 Cluster [18℄ observations indi ate. However, the dark energy puzzle 2

ould well involve a 'slight' modi ation of gravity, su h as the paradigmati s alar-

tensor theories of gravity  to be introdu ed later in (1.2.2) , where a new eld/s

would play the role of a dynami al dark energy (see [24℄ for a well-known modern

review on some proposals of s alar elds models of dark energy).

The equations of GR (1.1.1) an be derived 3

through a variational prin iple from

the Einstein-Hilbert a tion

S EH

^{, the} gravitationalpart, plus amatter part

S M

^:

0 = δS T ≡ δ(S ^EH + S M )

^(1.1.3)

with S EH ≡ − 1

2κ Z

M

√ g(R − 2Λ)

^(1.1.4)

2

The dark energy (or osmologi al onstant) problem an be divided into two: rst, why the

va uumenergywhi hbehavesasa osmologi al onstantisnotashighasitshouldbe,bymany

orders ofmagnitude,from theee tiveeld theorypointofview(netuning problem)and, se ond,

why isit sosmall that be omes dominant pre isely at thepresenttime ( oin iden eproblem). See

e.g.[62℄fora omprehensivereview.

3

AspointedoutinWald'sbook [80℄,Appendix E,thisderivation ignoresaboundarytermunless

onetakesto bezeronotonlythevariationsof themetri at theboundary,but alsotheirderivatives

(seealso[45℄).

where

M

^{is the spa etime manifold over whi h we integrate and}

g ≡ −det(g ^µν )

^{. The}

formulation in terms of an a tion is essential for a quantum approa h based on the

path-integral formalism. Moreover, it gives us an interesting perspe tive on GR, one

where symmetryplays a ardinalrole. Consider the groupof external transformations

given by the group of dieomorphismsinthe manifold,

Dif f (M)

^:

Active map : x ^µ −→ y ^µ = y ^µ (x)

^(1.1.5)

In passing,we willalways takethe `a tive view' when onsidering(gauge) symmetries

of the Lagrangian or the a tion and the `passive view' when we speak of oordinate

transformations or ovarian e of the equations; the former involves a map among the

points in the manifold, the latter a mere hange of oordinates. Under this transfor-

mation (1.1.5), the metri tensor behaves as:

g αβ (x) −→ g µν ^′ (x) = ∂x ^α

∂y ^µ

∂x ^β

∂y ^ν g αβ (x) − [g µν ^′ (x ^′ ) − g µν ^′ (x)]

whi h,at the innitesimal level

x ^µ −→ y ^µ = x ^µ + ξ ^µ (x)

^means

g µν (x) → g ^µν (x) + ∇ ^µ ξ ν (x) + ∇ ^ν ξ µ (x)

^(1.1.6)

(this hangeis just the Lie derivative). Now, letus write the most general a tionof a

se ond-ranktensor

g µν

^{symmetri under}

Dif f (M)

^{inan(innite)derivativeexpansion:}

S EF T = Z

M

√ g  1

κ Λ − 1

2κ R + c 1 R ² + c 2 R µν R ^µν + ... + L M



(1.1.7)

Here, ea h urvature  nomatter

R

^or

R µν

^{,or even}

R µνρσ

^ ontributestwo deriva- tives in the expansion (

∼ ∂ ² g µν

^{). GR} orresponds to the two lowest-order terms, the ontribution of the rest being irrelevant at ordinary s ales. It is in this pre ise sense

that one an speak of General Relativity as a (quantum) ee tive eld theory with a

Dif f (M)

^{symmetrygroup.}

One ould obje t that this external gauge symmetry, orthis general ovarian e of

the a tion, is physi ally va uous, following an argument by Kret hman that every

spa etime theory admits a generally ovariant representation [52℄. However, other

spa etime theories with just the dynami al entity

g µν

^{, the metri , do require the in-}

trodu tionof absoluteba kgroundobje tsinorderto a hievea ovariant formulation,

and in this pre ise sense GRis remarkably simple[64,65℄.

Thequantumtheorybasedonthe Einstein-Hilberta tion(1.1.4)ispower- ounting

not-renormalizable. This is due to the fa t that the oupling onstant

κ

^{arries di-}

mensions (of length squared), whi h for es the appearan e of two extra derivatives in

the numerator atea h onse utive order in a loop expansion. What it means is that,

startingattreelevelwith(1.1.4),oneexpe ts togenerate theinniteseriesof termsin

(1.1.7)by onsideringquantum orre tionswithanin reasingnumberofloops. Indeed,

expli it al ulationshaveestablishedthatGeneralRelativitywithmatter(without os-

mologi al onstant) generi ally diverges atone loop[74℄ through a term proportional

tothe square of the urvature,and pure gravity diverges attwo loops [46℄with aterm

ubi inthe urvature. Therefore, the theory has nopredi tivepower unlessone stays

withinthe realm ofanee tive eldtheory, validfor the lowenergy regimebelow

M p

^.

Several approa hesareenvisionedtoprovideanultraviolet(high-energy) ompletionof

the theory. Startingfromthe most onservative,one an hopeto nda non-trivialul-

travioletxed-pointintherenormalizationgroupowoftheinnitedimensionalspa e

of ouplings of (1.1.7), as rst advan ed by Weinberg [81℄; this has been named the

asymptoti safety s enario for gravity (see [61℄ fora review). Then,there isa re ent

proposalby Horava [48℄ (see also[16℄) that breaks Lorenzinvarian eat high energies,

thus involving a preferred foliationof spa e-time; this approa h is under vigorous ex-

amination at present. Many other ways to render gravity a fully-edged theory have

been proposed (see e.g. [34℄ for a list), in luding notably strings and loop quantum

gravity; they appear to involve somehowmore radi al assumptions.

Puttingasidethe aforementioned on ernsforafullquantum theoryofgravity,one

an always rea h a onsistent quantum theory by means of linearizing the equations

of General Relativity. In terms of the a tion, this involves expanding the metri in

a at Minkowski ba kground plus a perturbation,

g µν = η µν + h µν

^{, and keeping only}

quadrati terms in

h µν

^{, whi h orrespond to free} propagation without intera tions.

Indeed, indoingso,one obtainsthe Fierz-PauliLagrangian ofa freemasslessspin-two

parti le,the graviton[44℄:

L = L ^I + βL ^II + aL ^III + b L ^IV

^(1.1.8)

with

L ^I ≡ ¹ 4 ∂ _µ h ^νρ ∂ ^µ h _νρ , L ^II = − ¹ 2 ∂ _µ h ^µρ ∂ _ν h ^ν _ρ , L ^III = ¹ ₂ ∂ ^µ h∂ ^ρ h _µρ , L ^IV = − ¹ 4 ∂ _µ h∂ ^µ h

where

β = 1; b = (1 − 2a + 3a ² ) /2.

^{The freeparameter} orresponds toredenitionsof themetri eld( hangesofframe);a tually,theusualpresentationoftheLagrangianis

with

a = b = 1

^{(seee.g.[1℄). Thistheoryretainsasymmetryunderlinear}(innitesimal) dieomorphisms

h µν (x) → h ^µν (x) + ∂ µ ξ ν (x) + ∂ ν ξ µ (x),

^(1.1.9)

as ompared to (1.1.6). In Wald's words, the full theory of General Relativity may

be viewed as that of a massless spin-2 parti le whi h undergoes a nonlinear self-

intera tion.

However, this is not the end of the story. Conversely, one an bootstrap its way

ba k to General Relativity from the linearized theory in a natural way [17,32,42℄.

For a sket h of the argument, as well as the motivation for a spin-2 parti le to start

with, see e.g. [3℄

4

. In the same referen e, it is also outlined the a tual proof [17℄ that

used a Palatini formalism, with the metri and the ane onnexion as independent

variables. There has been some dis ussion regarding the assumptions that this proof

a tually involves(see [66℄ and the later reply [31℄). To the best of my knowledge, the

main issue is that one is able to re over the equations of motion of GR, but not the

omplete Einstein-Hilberta tion  whi h in ludessome total-divergen e terms.

At this point, there is a ru ial observation for our work. For a onsistent de-

s ription in at spa e of the massless spin-2 graviton the ne essary and su ient

gauge symmetry is the linear (innitesimal)

T Dif f (M)

^{group [78℄,}

x ^µ −→ y ^µ = x ^µ + ξ ^µ (x), with ∂ µ ξ ^µ (x) = 0

^{, with three} independent fun tionalparameters and a t- ing as

h µν (x) → h ^µν (x) + ∂ µ ξ ν (x) + ∂ ν ξ µ (x), with ∂ µ ξ ^µ (x) = 0,

^(1.1.10)

as opposed to the full

Dif f (M)

^{a ting inexpression (1.1.9). The derivation involves}

the little group of Lorentz transformations for a massless parti le of momentum

k ^µ

^,

with

k ² = 0

^{. This grouphasthree}generators, beingisomorphi tothe ISO(2)groupof translationsand rotationsintheplane. Lookingforthe unitaryrepresentations ofthis

generators  whose eigenstates orrespond to the dierent polarizationstates of the

parti leforagiven

k ^µ

^{,onenoti esthatthe(unitary)}representationsofthetwonon- ompa ttranslation generatorsareinnitedimensional,leadingtoaninnitenumber

of polarizations, whi h is totally unphysi al. This atastrophi degenera y is avoided

by demanding that these generators a t trivially, and de laring the equivalen e of all

4

There is a subtlety in the onstru tion: the (self) energy-momentum tensor for gravity in at

spa ethat onehastouseisnottheonegivenbytheusualBelinfantepres ription[31,66℄.

these polarizations, whi h are then related by a gauge transformation. For a spin-2

parti le,toberepresentedbyarank-twotensor

h µν

^{,oneisdire tlyledto(1.1.10). Note}

in parti ular that the tra e

h ≡ h ^µν η ^µν

^{is aLorentzinvariant and thus we an restri t}

ourselvesto gaugesymmetrieswhi h leave itinvariant:

∂ µ ξ ^µ = 0

^{amountspre isely to}

this.

The work to be exposed in this thesis is based upon this observation, and the

hypothesisthat TDiis the truesymmetry that des ribes gravity in nature,asopposed

to Di. That is to say, one may well think of the non-linear realizations of TDi

symmetry, whi h would be well-motivated theories from a quantum perspe tive 

sin e the linearized theory in at spa e in ludes the graviton just the same , and

whi h ouldpossiblyyieldasolutiontotheshort omingsofGR,essentiallywithregard

to darkenergy.

Inordertodevelopthisprogram,onerstneedsanexhaustiveknowledgeofthe'zo-

ology'of linear Lagrangians,based on

h µν

^{. This wasdone in[1℄. Thesame on lusion}

that

T Dif f (M)

^{is the ne essary and su ient symmetry for a onsistent} des ription ofthe masslessgravitonwasarrivedthrough the requirementsof lassi alstabilityand

absen e of ghosts. The most generalLagrangian reads:

L = L ^I + βL ^II + aL ^III + b L ^IV

^(1.1.11)

with

L ^I ≡ ¹ 4 ∂ _µ h ^νρ ∂ ^µ h _νρ , L ^II = − ¹ 2 ∂ _µ h ^µρ ∂ _ν h ^ν _ρ , L ^III = ¹ ₂ ∂ ^µ h∂ ^ρ h _µρ , L ^IV = − ¹ 4 ∂ _µ h∂ ^µ h

where

β = 1

^{, while}

b

^and

a

^{are arbitrary; ompare with (1.1.8). This symmetry an}

be enhan ed in two distin t ways: one possibility is of ourse to onsider the full Di

group, the other (denoted WTDi) adds a Weyl symmetry, by whi h the Lagrangian

be omes independent of the tra e.

Then, in the se ond pla e, from the linear Lagrangian (1.1.11) one an hope to

nd the non-linear extension in a systemati way, through the analogous of the pro-

edure in [17,32℄ (the one that bootstraps GR from the Fierz-Pauli Lagrangian); this

was attempted in [13℄ (see also [14℄, with profuse information on TDi theories). It

is simpler,however, to relyonthe non-lineargeneralizationof the symmetry transfor-

mation (1.1.10) in order to onstru t dire tly the non-linear Lagrangian (same refer-

en es[13,14℄). Asamatteroffa t,one an onsistentlykeepthesamerestri tiononthe

innitesimal parameter of the transformation 5

,

∂ µ ξ ^µ (x) = 0

^{, leading to the non-linear}

(innitesimal)form:

g µν (x) → g ^µν (x) + ∇ ^µ ξ ν (x) + ∇ ^ν ξ µ (x), with ∂ µ ξ ^µ (x) = 0,

^(1.1.12)

and thus,

pg(x) −→ pg(x) + ξ ^λ (x)∂ λ pg(x).

^(1.1.13)

For a nite transformation, this orresponds to the subgroup of (nite) general oor-

dinate transformations with Ja obian determinant equal to unity. Therefore, s alar

and s alar densities  su h as the determinant of the metri have the same trans-

formation properties as regards the

T Dif f (M)

^{group. As a} onsequen e, the TDi

ounterpartoftheEinstein-Hilberta tion(1.1.4)willin ludearbitraryfun tionsofthe

determinantof the metri , aswellas the usual terms:

S

TDi

= − 1 2κ ²

Z

d ⁴ x √ g



f (g) R + 2f λ (g) Λ + 1

2 f k (g) g ^µν ∂ µ g∂ ν g



+ S M

^(1.1.14)

and the mattera tion may be taken tobe of the form

S M = Z

d ⁴ x √ gL SM [ψ m , g µν ; g]

^(1.1.15)

where we allow for an arbitrary extra-dependen e on

g

^{in the} onventional Standard Model Lagrangian

6

. As it stands, the a tion given by (1.1.14), (1.1.15) is learly

non- ovariant, but is formulated in a spe i set of oordinates  up to TDi trans-

formations of oordinates.

A spe ial parti ular ase of these TDi theories omes from the non-linear gener-

alization of the WTDi linear ase, mentioned before. The restri tion on the tra e

there,manifestsitselfhere intheformofa onstrainonthevalueofthedeterminantof

the metri ,typi allyset to

√ g = 1

^{. For this reason, this theory is named unimodular}

gravity:

5

Theother mostsensible generalization,where

ξ ^µ

^{is restri tedwith}

∇ µ ξ ^µ = 0

^(and

g

^doesnot

transformat all,notevenasas alardoes) isdoomedsin eone annot a ommodateakineti term

for

g

^:

¹ ₂ f k (g)g ^µν ∂ µ g∂ ν g

^.

6

Forexample,theEle tromagneti Lagrangianisgeneralizedto

− ¹ 4 f _EM (g)F µν F ^µν

^,with

f _EM

^an

arbitraryfun tion.

S U G [ˆ g µν ] ≡ − 1 2κ ²

Z

d ⁴ xǫ 0 R[ˆ g µν ], with det(ˆ g µν ) = 1

^(1.1.16)

This was the histori approa h for a non-linear realization of TDi, from the seminal

paper[78℄. A tually,Einsteinhimselfplayed withthistypeof theorybeforedeveloping

GeneralRelativity(see 14 in[67℄). Wewillnot besointerestedinunimodulartheory

be ause itis relatively well studied (a sele tionof referen es: [4,38,60,73,76,78℄) and

does not oer many opportunities for phenomenology.

The study of the quantum feasibility of gravity theories based on the TDi sym-

metry  ex luding UV behavior  was performed in [37,53℄ (see also [2℄). For this

purpose, aBRST analysis was employed; BRST is a powerful symmetry of the quan-

tumLagrangian,inthepresen eofthegauge-xingandghosttermswhi hhavebroken

the usual gauge symmetry [12℄. One di ulty that appears isthe presen e of  hains

of ghosts, orghosts for ghosts. Also, the gauge-xing introdu es higher derivatives in

the a tion,whi h inturn produ einfrareddivergen es; this problemista kledwith by

introdu ingadditionalelds [53℄, proving infrared-niteness to allorders of perturba-

tion theory. The on lusion isthat TDi is aperfe tlysuited gaugegroup todes ribe

a unitary theory of gravitation.

1.2 Outline

Therst questionwedeal withisthe ultravioletbehaviorofTDitheories[8℄,in hap-

ter 2. Havingintoa ountthefoundationalquantuminspiration forthesetheories, it

is naturaltond out whether thesame divergen es appear thaninGeneralRelativity.

The 1-loop divergent pie es were al ulated using the ba kground eld method and

heat kernel te hniques. Theba kground eldmethodsplitsthe quantum eld

g µν

^into

a ba kground

¯ g µν

^{plus a} perturbation

δg µν

^{. It is espe ially useful in gauge theories}

sin e it retains a ertain ba kground ovarian e inspite of xing the gauge. It is on-

venient tomake the ba kground

g ¯ µν

^{fulllthe lassi alequationsof motionand, then,}

the 1-loopee tivea tion an be al ulated through the formula:

Γ ⁽¹⁾ = S 0 + i

2 Ln [det(O)]

^(1.2.1)

where

S 0

^{is the lassi al a tion and}

O

^{is the operator quadrati in the} perturbations

δg µν

^{, whi h in ludes} ontributions from the gauge-xing and Faddeev-Popov ghosts terms; all quantities depend only on the ba kground. It is to al ulate the divergent

part of the determinant of the operator

O

^{that one resorts to} heat-kernel te hniques.

Theexpressionsforso- alledminimaloperatorsaretabulated. Oneofthemainissues

for this al ulation was to devise a lever gauge in whi h the operator a quired this

minimal form. This was a hieved through the introdu tion of a Stue kelberg eld:

a 'tri k', aspurious eld that enlarges the gauge symmetry. In this way, we basi ally

endedup al ulatingdivergen esofas alar-tensortheory,whi hwastherstindi ation

of a ertainequivalen e with it.

In the se ond pla e, we address the observational viability of TDi theories from

its denition (1.1.14)(1.1.15) [9℄, in hapter 3. As was stated in the introdu tion,

there is an in redible amount of data that supports General Relativity. The aimis to

identify the orner of (fun tional) parameter spa e for TDi that is ompatible with

observations, and whether there is onsiderable freedom to go beyond GR or we are

basi ally onstrained to small deviations from it. We took into onsideration a wide

variety of tests, in luding solar system tests, binary pulsars, osmologi al evolution

or deviations from Newton's law (fth for e). Spe ially relevant were the tests of

the universality of free fall, UFF (

m inertial = m passive

^{) [83℄, Newton's third law (or}

momentum onservation) [63℄ and lo al position invarian e, LPI [77℄, sin e all these

are extremely onstraining. We also took into a ount the similarity with the well-

studied s alar-tensor theories, to be presented below. On the other hand, we did not

dwell into a possible MOND (Modied Newtonian Dynami s) phenomenology with

TDi with a view on the Dark Matter problem, sin e it did not seem to have good

prospe ts [20℄.

Finally, in hapter4 we put together several pie es of a small puzzle and establish

a rigorous orresponden e between TDigravity and s alar-tensortheories, analogous

to the relation of unimodular gravity with General Relativity [56℄. In s alar-tensor

theory there is a s alardegree of freedom

φ

a ompanyingthe metri tensor:

S ST = − Z

M

√ g(f (φ)R − 2f ^λ (φ)Λ − 1

2 g ^µν ∂ µ φ∂ ν φ) + S M [g µν , φ]

^(1.2.2)

(In passing, an obvious generalization are multi-s alar-tensor theories). This s alar

appears asanatural onsequen e inmany theoriesBeyond the Standard Model and,

inparti ular, onstitutesthe ommonme hanismadvo atedtoprodu eaninationary

epo hin the universe. One an he k that indeedthe stru ture of (1.1.14) and (1.2.2)

are remarkably similar. This equivalen e is of mu h use, as we will on lude, sin e

s alar-tensor phenomenologyis ex ellently studied.

Ultraviolet behavior of TDi gravity

2.1 Introdu tion

There are interesting theories of gravitation, su h as Einstein's 1919 tra eless one,

whi h have the pe uliarity that in order to obtain the eld equations from an a tion

prin iple, the allowed variations

δ ^t g µν

^{must obey}

g ^µν δ ^t g µν = 0

^{. This parti ular model}

has attra ted some attention sin e it provides a dierent point of view on the os-

mologi al onstant problem, in the sense that from Einstein's tra eless equations one

an obtain the usual equationsof GeneralRelativity withthe addition ofan arbitrary

integration onstantthat playsthe roleof the osmologi al onstant (in[1,4℄ageneral

referen e an be found).

Arelatedslightlymoredrasti possibilityistorestri tthesymmetryofnaturetothe

subgroupin ludingthosedieomorphismsthatenjoyunitJa obiandeterminant,whi h

weshalldubtransverse 1

orTDi. Thissubgroupisinterestingbe auseifthespa etime

symmetryisTDi,thennaturemakesnodistin tionsbetweentensordensities: theyall

transformastruetensors. Inparti ular,thedeterminantofthemetri isas alarunder

TDi,andwehavefreedomtoin ludeoperatorsinoura tionsexa tlyasonedoeswith

usual matters alars. In arelatedvein, we have re entlyproposed toymodels inwhi h

not onlythe va uumenergy,but every formof potentialenergy doesnot gravitate [7℄,

1

We areof ourseawareof thefa t that not everydieomorphism lose to theidentity lieson a

oner-parametersubgroup[33℄.

althoughtheremay besome subtletiesinthe ouplingbetween matterand gravity [6℄.

Transverse higherspin theorieshavebeen analyzed in[15,72℄.

Theaimofthis workistostudy transversegravity,that is,gravitymodelsenjoying

this restri ted symmetry prin iple, to one-loop order. For one reason: it has been

shown that in many ases transverse theories are equivalent, at the lassi al level, to

ordinary gravity with an extra s alar parti le in the eld ontent, akin to the usual

dilaton, and with an arbitrary osmologi al onstant. This has been studied in some

detail in the se ond referen e in [1,4℄. As far as we know, quantum ee ts may spoil

the equivalen e. Furthermore, there are grounds to suspe t that transverse theories

ought to enjoy better ultravioletbehavior than Einstein'sgravity, be ause there is no

divergen e asso iated to the onformal mode. On the other hand, there is no reason

why this should be seen in perturbation theory. We shall begin by dis ussing the

pe uliarities of this kind of models with respe t to General Relativity (GR) and the

orresponding problem of performing a al ulation where one annot dene a simple

ovariantgauge-xing,andthenwewillstudyanequivalents alar-tensortheorywhi h

is the one we eventually ompute to one looporder.

2.2 The one-loop transverse ee tive a tion

The prin ipalhypothesiswewilladoptinthispaperisthat thespa etimesymmetryof

natureisnotthefullsetofarbitrary oordinate hanges,inthesensethatitisassumed

in Einstein's General Relativity, but only the subgroup of dieomorphisms su h that

the determinant of the orresponding Ja obian equals unity. On e we assume this

symmetry prin iple, a ouple of importantdieren es with respe t toGR arise.

The rst one of ourse is that now one is not able to distinguish between tensor

densitiesof dierentweight. Atensordensityis anobje tthat underana tive hange

of oordinates (Di)

x ^µ → y ^µ (x)

^(2.2.1)

transforms as

T ^′ν _µ ¹ ₁ ^...ν _...µ ^l _n (y) = [D(y, x)] ^ω ∂x ^{ρ 1}

∂y ^{µ 1} . . . ∂x ^{ρ n}

∂y ^{µ n}

∂y ^{ν 1}

∂x ^{σ 1} . . . ∂y ^{ν l}

∂x ^{σ l} T _ρ ^σ ₁ ¹ _...ρ ^...σ _n ^l (x)

^(2.2.2)

where

ω

^{is the weight of the density and we have denoted}

D(y, x) ≡ det  ∂y ^µ

∂x ^ν



(2.2.3)

i.e., the determinant of the Ja obian. It is plain that were we torestri t our transfor-

mations tothose that obey

D(y, x) = 1,

^(2.2.4)

theneverydensitybehavesasatensor. Themostimportant onsequen eofthisasump-

tion is that two ru ial s alar densities of GR, the determinant of the metri that

represents the dynami s of gravity, as well as the integration element

d ⁿ x

^{, are now a}

true s alar and dual to a true s alar respe tively. Therefore, we are free to use the

determinant of the metri in the same way as any other s alar in the theory, writing

down operatorsthat were forbiddenby the symmetry before.

It has been shown in the se ond referen e of [1,4℄ that at the linear level models

invariant under (the linealizationof)TDi propagate an additionaldegreeof freedom

in luded inthe metri besides the usual spin two graviton. Eventually this mode will

be responsible foran important pie e of the divergen es.

Se ondly, in GR arbitrary hanges of oordinates are onsidered as a gauge sym-

metry that, as usual, one must x. On a manifold of dimension

n

^{, there are}

n

^gauge

onditionsoneshouldgivetogaugexthelo alsymmetry. Then,thereisenoughroom

todothe xing ina ovariantway 2

,whi h isvery usefulto simplify omputations. An

example of gauge ommonlyused is the harmoni (orminimal orDeWitt) gauge

χ ν ≡ ¯ ∇ ^µ h µν − 1

2 ∇ ¯ ^ν h = 0

^(2.2.5)

where

h µν

^{isthegraviton}u tuation,

h

^{itstra ewithrespe ttotheba kgroundmetri}

and the ovariant derivativesare onstru ted withthe sameba kground metri . Now,

a slightly smaller symmetry means also less gauge onditions to x. In parti ular,

(2.2.4) for es one of the

n

^{original gaugeparameters to be determinedin terms of the}

others in su h a way that there are only

n − 1

^{gauge onditions to spe ify, resulting}

in the impossibility to rea h a onvenient ovariant gauge like the previous one. Of

2

Hereandinwhat followswearereferingto ovarian ewithrespe tto theba kgroundsymmetry

maintainedintheBa kgroundFieldmethod,andunderwhi h(2.2.5)isave tor. Asitiswellknown,

thegaugexingtermmustbreakthequantumsymmetry(2.5.27).

ourse, it is always possible to nd, instead of a ve tor that vanishes and givesus the

desired

n

onditions,

n − 1

^{s alars} onstru ted outofthe gravitonu tuation,itstra e and derivatives like for example

χ 1 ≡ ¯ ∇ ^µ ∇ ¯ ^ν h µν , χ 2 ≡ ¯ ∇ ² h , . . .

^(2.2.6)

Thevanishingofthese

χ 1 , . . . , χ n −1

^{s alars} onstitutesana eptable olle tionofgauge onditions to x the TDi symmetry of the system. Another possibility mentioned

in [1,4℄is toproje t the harmoni gauge into the transverse dire tion

χ ^t _ν ≡ ¯ ∇ ^ν ∇ ¯ ^ρ ∇ ¯ ^σ h ρσ − ¯ ∇ ² ∇ ¯ ^µ h µν

^(2.2.7)

giving automati ally

n − 1

independent onditions. Both gauge xing hoi es, even if perfe tly valid from a gauge theory point of view, are not suitable to undertake a

al ulation, the reason being that in general the operator obtained for the graviton

u tuations (and in identallyfor the ghosts) does not take a minimalform, in a very

pre ise sense. Inparti ular, it annotbe put inthe form ofa Lapla ian(see (2.5.19)).

Everyonethathasworkedoutaone-loop omputationusingBa kgroundFieldmethods

andHeatKernelte hniquesmayappre iatethedi ultiesindealingwithnonminimal

operators, thoughthere are known tra table examples [11℄.

Toavoidthisunne essary ompli ationwewillintrodu ea ompensatoreld(some

sort ofStue kelberg eld)thatrenders thetheory Diinvariantand sothatwere over

the original model in the, so to say, unitary gauge" (in analogy with the breaking

of Ele troweak symmetry) in whi h the ompensator disappears from the spe trum.

Imposingthispartialgauge isoneof the

n

^{onditions ofthe Diinvarian eand weare}

left with the

n − 1

^{onditions of TDi, as it should be. The tri k lies in} maintaining the full invarian e duringthe al ulation inorder to obtain a minimaloperator xing

the gauge as in standard GR, but the pri e to pay is that the ompensator will not

vanish,sin ewe have nosymmetrylefttorea hthe unitarygauge, andwillbepresent

in the nal result.

Let us start with a parti ular example of a TDi a tion, whi h is not the most

general one an give. Consider the a tion

S g = − 1 2κ ²

Z

d ⁿ x √

g ^∗ [f (g ^∗ ) R ^∗ + 2f λ (g ^∗ ) Λ]

^(2.2.8)

where

Λ

^{plays the role of a} osmologi al onstant,

f

^and

f λ

^{are arbitrary fun tions}

of the determinant of the metri

g ^∗ ≡ det g µν ^∗

^{, and the a tion is in general not Di}

invariant, ex ept in the trivial ase inwhi h

f

^and

f λ

^{are onstants. Moreover, under}

a Dithe a tion transforms to

S g = − 1 2κ ²

Z

d ⁿ x √

g ^∗ f(g ^∗ C ² ) R ^∗ + 2f λ (g ^∗ C ² ) Λ 

(2.2.9)

where

C(x)

^{is a} ompensator eld, dened so that

ϕ ^∗ ≡ g ^∗ C ²

^{transforms as a true}

s alar (see(2.2.3)). Theaforementionedunitarygauge would orrespond tothe hoi e

C = 1

^{, re overing the original a tion. We an write in terms of the s alar eld a}

perfe tly Diinvarianta tion

S g = − 1 2κ ²

Z

d ⁿ x √

g ^∗ [f (ϕ ^∗ ) R ^∗ + 2f λ (ϕ ^∗ ) Λ]

^(2.2.10)

To perform the omputation is onvenient to go to the Einstein frame, so we make a

onformal transformation

g µν = Ω ² g _µν ^∗ g = Ω ²ⁿ g ^∗

ϕ = gC ² = Ω ²ⁿ g ^∗ C ² = Ω ²ⁿ ϕ ^∗

^(2.2.11)

If we hoose the onformalfa tor (supposing

n 6= 2

^)as

Ω ^{n −2} = f (ϕ ^∗ ) = f (Ω ⁻²ⁿ ϕ)

^(2.2.12)

then in termsof the new metri the a tiontakesthe form

S g = − 1 2κ ²

Z

d ⁿ x √ g [R + 2F λ (Ω) Λ] + (n − 1)(n − 2) 2κ ²

Z

d ⁿ x √ g 1

Ω ² g ^µν ∂ µ Ω∂ ν Ω

(2.2.13)

where we have made use of (2.2.12) inorder to express

f λ

^{in terms of}

Ω

Ω ⁻ⁿ f λ (Ω ⁻²ⁿ ϕ(Ω)) ≡ F ^λ (Ω)

^(2.2.14)

Noti e howeverthatthis reasoning annotbeappliedwhen

f (g ^∗ ) = g ^{∗ 2−n 2n}

^{sin einthat}

ase

Z

d ⁿ x √

g ^∗ f (ϕ ^∗ ) R ^∗ = Z

d ⁿ x √

g f (ϕ) R

^(2.2.15)

and one annotget tothe Einsteinframe. Asimilarproblemarisesif

f (g ^∗ ) = constant

sin e then (2.2.12)is not invertible to give

ϕ ^∗ = f ⁻¹ (Ω ^{n −2}

^{) or, inother words, we are}

alreadystartingintheEinsteinframeandthe onformaltransformationisnotdened.

Appart from these subtleties, anal redenition of the s alar

φ ≡ p2(n − 1)(n − 2) ln Ω

^(2.2.16)

givesus the desired a tion

S g = − 1 2κ ²

Z

d ⁿ x √ g [R + 2F λ (φ) Λ] + 1 2κ ²

Z

d ⁿ x √ g 1

2 g ^µν ∂ µ φ∂ ν φ.

^(2.2.17)

We have maintained the notation

F λ (φ)

^for

F λ (Ω (φ))

^{in the hope it will ause no}

onfusion.

In this form of the a tion the additional s alar degree of freedom is manifest, and

it is suitable for performing the al ulation using well known standard Ba kground

Field methods that, though straightforward, are quite tedious. The heavy details of

the omputationhavebeen relegated toanappendix. Thenal resultfor theone-loop

ountertermof the theory (2.2.8),in termsof the originalvariables, reads

∆S = 1 ǫ

1 (4π) ²

Z

d ⁴ x √

g ^∗  1827

160 f ⁻⁴ f ^′4 (g _∗ ^µν ∂ µ ϕ ^∗ ∂ ν ϕ ^∗ ) ² + 171

20 Λ f ⁻³ f ^′2 f λ g _∗ ^µν ∂ µ ϕ ^∗ ∂ ν ϕ ^∗ − 57

5 Λ ² f ⁻² f _λ ² + 1

9 Λ ² f ^′−1 f _λ ^′ − 2f ⁻¹ f λ

 2

+ 2

9 Λ ² f ² 4 f ⁻² f λ − 3f ⁻¹ f ^′−1 f _λ ^′ + f ^′−2 f _λ ^′′ − f ^′−3 f ^′′ f _λ ^′ 

× 2 f ⁻² f λ − 3f ⁻¹ f ^′−1 f _λ ^′ + f ^′−2 f _λ ^′′ − f ^′−3 f ^′′ f _λ ^′ 



(2.2.18)

Primedenotes derivativewithrespe tto

ϕ ^∗

^{. Itis learthatour herishedhopethat,in}

the absen eof the onformalmode,the ultravioletbehavioroftransverse models ould

bebetter than the orrespondent inGR is not fullled. Even if the osmologi al on-

stantvanishes,the rst termin the ountertermremains,ex ept in ase

f

^{is onstant,}

but that orrespondsexa tlytotheEinsteinHilberta tion,whi hisknowntobeone-

loopnite [74℄. In fa t, the form of the ounterterm reminds the one obtained when

a s alar eld, possibly with a potential term, is oupled to gravity. That is be ause

the mode responsible for the divergen es is the additional mode in the metri whi h

annot be killedin the la k of full Diinvarian e,aswill be ome more transparent in

what follows.

Weshouldmentionhere thattheGR limit

f ^′ → 0

^{isnot regularifthe} osmologi al onstant does not vanish. But remember that this limit is one of the problemati

asesregardingthe onformaltransformation(seethe omentsfollowing(2.3.6)). Also,

quantum orre tionstend togenerate akineti energy term forthe determinantof the

metri ,soitis onvenienttoin ludeitinthebarea tionfromthe beginning,obtaining

a more omplete model.

2.3 A more general transverse a tion

Taking into a ount the last onsiderations, in this se tion we will extend the model

by introdu inga kineti energy term for the determinant of the metri . The resulting

a tion willbe the most general gravitatoryTDi a tionwith the usualproperties one

imposestoasuitablea tion(tobeas alarofthesymmetry,se ondorderinderivatives

et .) 3

S = − 1 2κ ²

Z

d ⁿ x √ g ^∗



f (g ^∗ )R ^∗ + 2f λ (g ^∗ )Λ + 1

2 f φ (g ^∗ )g _∗ ^µν ∂ µ g ^∗ ∂ ν g ^∗



(2.3.3)

so that, asbefore, afteran arbitrary hange of oordinates

S = − 1 2κ ²

Z

d ⁿ x √ g ^∗



f (ϕ ^∗ )R ^∗ + 2f λ (ϕ ^∗ )Λ + 1

2 f φ (ϕ ^∗ )g _∗ ^µν ∂ µ ϕ ^∗ ∂ ν ϕ ^∗



(2.3.4)

where the s alar eld is

ϕ ^∗ ≡ g ^∗ C ²

^{. We should now goto the Einstein frame through}

a onformaltransformation

g µν = Ω ² g _µν ^∗

^(2.3.5)

Choosing the onformal fa tor as

Ω ^{n −2} = f (ϕ ^∗ )

^(2.3.6)

3

One ouldhavein ludedapotentialterm

S V = − 1 2κ ²

Z

d ⁿ x √

g ^∗ M ² V (g ^∗ )

^(2.3.1)

but it anbeabsorbedinthedenition of

F λ (Ω)

^,i.e.,

2ΛF λ (Ω) ≡ Ω ⁻ⁿ 2Λf λ (f ⁻¹ (Ω ⁿ⁻² ) + M ² V (f ⁻¹ (Ω ⁿ⁻² )

(2.3.2)

so itdoesnotin ludeanyinterestingnewissueandwewon't onsiderit.

the a tion in the new frametakes the form

S = − 1 2κ ²

Z

d ⁿ x √ g [R + 2F λ (Ω)Λ] + 1 2κ ²

Z

d ⁿ x √ g  2(n − 1)(n − 2) Ω ²

−Ω ²⁻ⁿ f φ f ⁻¹ (Ω ^{n −2} )
 ∂f ⁻¹ (Ω ^{n −2} )

∂Ω

 2 # 1

2 g ^µν ∂ µ Ω∂ ν Ω

^(2.3.7)

where we have dened

F λ (Ω) ≡ Ω ⁻ⁿ f λ f ⁻¹ (Ω ^{n −2} )

(2.3.8)

A nal redenition of the s alar gives the desired a tion studiedearlier

"

2(n − 1)(n − 2)

Ω ² − Ω ²⁻ⁿ f φ f ⁻¹ (Ω ^{n −2} )
 ∂f ⁻¹ (Ω ^{n −2} )

∂Ω

 2 #

g ^µν ∂ µ Ω∂ ν Ω = g ^µν ∂ µ φ∂ ν φ

(2.3.9)

and onsequently we an use the ounterterm quoted in the appendix. We have xed

the sign of the kineti term of the new eld

φ

^{so that it is not a ghost, and then we}

are for ed torequirethe fun tion ofthe lefthand side tobepositivedenite. Interms

of the original fun tions itmeans

2(n − 1) f ^{2−n 2} − (n − 2) f ^φ f ^{n−4 n−2} f ^′−2 ≥ 0

^(2.3.10)

Finally,we are able towrite the one-loop ounterterm of the theory (2.3.3)

∆S = 1 ǫ

1 (4π) ²

Z

d ⁴ x √

g ^∗  203

160 3f ⁻² f ^′2 − f ⁻¹ f φ

 2

(g ^µν _∗ ∂ µ ϕ ^∗ ∂ ν ϕ ^∗ ) ² + 57

20 Λ 3f ⁻³ f ^′2 f λ − f ⁻² f λ f φ  g _∗ ^µν ∂ µ ϕ ^∗ ∂ ν ϕ ^∗ − 57

5 Λ ² f ⁻² f _λ ² + 1

3 Λ ² f ^′−1 f _λ ^′ − 2f ⁻¹ f λ

 2

3 − f f ^′−2 f φ

 −1

+ 1

2 Λ ² 3f ⁻¹ − f ^′−2 f φ

 −4

× 24f ⁻³ f λ − 18f ⁻² f ^′−1 f _λ ^′ − 6f ⁻¹ f ^′−3 f ^′′ f _λ ^′ + 6f ⁻¹ f ^′−2 f _λ ^′′ − 10f ⁻² f ^′−2 f λ f φ

+7f ⁻¹ f ^′−3 f _λ ^′ f φ − 2f ⁻¹ f ^′−3 f λ f _φ ^′ + 4f ⁻¹ f ^′−4 f ^′′ f λ f φ − 2f ^′−4 f _λ ^′′ f φ + f ^′−4 f _λ ^′ f _φ ^′ 

× 12f ⁻³ f λ − 18f ⁻² f ^′−1 f _λ ^′ − 6f ⁻¹ f ^′−3 f ^′′ f _λ ^′ + 6f ⁻¹ f ^′−2 f _λ ^′′ − 2f ⁻² f ^′−2 f λ f φ

+7f ⁻¹ f ^′−3 f _λ ^′ f φ − 2f ⁻¹ f ^′−3 f λ f _φ ^′ + 4f ⁻¹ f ^′−4 f ^′′ f λ f φ − 2f ^′−4 f _λ ^′′ f φ + f ^′−4 f _λ ^′ f _φ ^′

− 4

3 f ⁻¹ f ^′−4 f λ f _φ ²



(2.3.11)

Let usremark that when

Λ = 0

^{and the fun tionsin front of the kineti term and the}

EinsteinHilbert term are

f = f φ = 1

^{we re over the result of 't Hooft and Veltman}

for gravity oupledtoa s alar withoutpotential

∆S = 1 ǫ

1 (4π) ²

Z

d ⁴ x √ g ^∗ 203

160 (g _∗ ^µν ∂ µ ϕ ^∗ ∂ ν ϕ ^∗ ) ²

= 1 ǫ

1 (4π) ²

Z

d ⁴ x √ g ^∗ 203

40 R ^∗2

^(2.3.12)

Noti e in passing that now the limit

f ^′ → 0

^{isnot singular, and this is due to the}

presen e of akineti energy termfor thes alar even if the onformaltransformation is

not dened. Moreover, the following diferentialequation relatingboth fun tions

2(n − 1)f ⁻¹ f ^′2 − (n − 2)f ^φ = 0

^(2.3.13)

saturatesthe bound(2.3.10)andhasarealsolutioniftheprodu t

f f φ

^{,asafun tionof}

the determinantofthemetri ,ispositivedenite, andthereforethereisanotherfamily

of one-loopnite theories in ase

Λ = 0

^. Nevertheless, after aneasy omputation one may prove that given the a tion (2.3.3), and under the hypothesis that the arbitrary

fun tionsverify(2.3.13),it(almost)alwaysexistsa onformaltransformation,whi his

pre iselly (2.3.6), that leads tothe EinsteinHilbert a tion. As a result, the family of

theories(2.3.13)arenothingbutGRwritteninanotherframe,withfullDiinvarian e.

Under this pointof view the one-loopniteness is not surprising.

The onlytheory that annot be put inEinsteinHilbert form is pre iselywhen

f (g ^∗ ) = g ^{∗ 2−n 2n}

^(2.3.14)

and

f φ

^{isgiven by(2.3.13). It anbeseenthatthis theoryhasanadditionallo alWeyl}

symmetry

g µν → Ω ² (x)g µν

^(2.3.15)

that prevents us from going to the Einstein frame. In fa t, this theory is exa tly the

WTDi modelof these ondreferen ein[1,4℄,whi hisaunimodular"model(that is,

a theory that an be written in terms of a metri with unit determinant and nothing

else) but writtenin termsof a metri not restri ted tohave unit determinant.

It is important to mention that both nite transverse theories have an enhan ed

symmetrythatallowsustoremovefromthe spe trumtheadditionaldegreeoffreedom

ontained in the metri . Then, as we have repeatedly advertised, the observed worse

behavior inthe ultravioletis due to this mode.

2.4 Con lusions

The main on lusionof our investigationis thatthere are onlytwo transverse theories

of gravity that are nite on shell. The rst one appears when TDi is enhan ed to

Di (and besides, the osmologi al onstant is ne tuned to zero); that is Einstein's

gravity, whoseonshellone-loopniteness wasproven ina lassi work by'tHooftand

Veltman[74℄.

The other theory enjoys also a greatersymmetry, a lo alWeyl invarian e denoted

WTDi, that allows to remove the additional degree of freedom present in generi

transverse models. One ould then be sure that the divergen e found is pre isely due

to this mode. WTDi theories in lude the so- alled unimodular ones, whi h an be

written using only the metri

g ˆ µν

^{su h that}

det ˆ g µν = 1

^{, but the lass of WTDi}

ould perhaps be larger that the unimodular one. Here we should mention that the

omputationwasdone inthe Einsteinframe, inwhi hthere isnofun tion(otherthan

thesquarerootofthedeterminant)infrontofthe urvatures alar. However,aswehave

said, te hni ally is not possible to rea h this frame in a theory with Weyl invarian e

like WTDi, for obvious reasons. Therefore, though the model with WTDi veries

(2.3.13),itfallsintothe aseswe annottreatwithourformalism. Stri tlyspeakingone

should then repeat the al ulationin anarbitraryframeto be sureof the on lusions,

but attheend the resultwill ertanly bethesame sin eitsphysi aloriginseemstobe

lear: the absen e of the s alar mode.

It should be remarked also that we a tually have al ulated in a Di invariant

theory whi h oin ides with the transverse theory ofour interest inthe unitarygauge

C = 1

^{. Our} omputation was done in the equivalent of the renormalizable gauge for YangMills theories, and it does ultimately rely on gauge invarian e of the extended

theory. In this sense it would be interesting to extend the analysis of the existen e

of a nilpotent BRS symmetry perhaps alongthe lines of what wasdone for transverse

theories in[37,53℄.

2.5 Appendix: te hni al notes

2.5.1 Some details of the omputations

To begin, let usbe quiteexpli it on our notationand onventions.

The at tangent metri is mostly negative

η ab ≡ diag (1, −1, −1, −1) .

^(2.5.1)

The Riemann tensor is

R ^µ ναβ ≡ ∂ ^α Γ ^µ _νβ − ∂ ^β Γ ^µ _να + Γ ^µ _σα Γ ^σ _νβ − Γ ^µ σβ Γ ^σ _να

^(2.5.2)

and we dene the Ri itensor as

R µν ≡ R ^{λ µλν} .

^(2.5.3)

Our onventions for the osmologi al onstant are su h that for a onstant urvature

spa e

R µν = − 2

n − 2 λg µν

^(2.5.4)

then the ordinary de Sitter spa e has negative onstant urvature, but enjoys positive

osmologi al onstant. The EinsteinHilbert a tion is onsequently dened as

S = − c ³ 2κ ²

Z

d ⁿ x p|g| (R + 2λ) + S ^matter

^(2.5.5)

with

κ ² ≡ 8πG

^.

Ba kground ovariant derivatives an be integrated by parts:

Z

d ⁿ x p|¯g| ¯∇ ^µ L ^µ = Z

d ⁿ x p|¯g| 1 p|¯g| ∂ µ

p |¯g|L ^µ 

= Z

d ⁿ x ∂ µ

p |¯g|L ^µ 

(2.5.6)

and some useful ommutators with our onventios are:

 ∇ ¯ ^β , ¯ ∇ ^γ  ω ρ = ω µ R ¯ ^µ ργβ

 ∇ ¯ ^β , ¯ ∇ ^γ  V ^ρ = −V ^µ R ¯ ^ρ µγβ

 ∇ ¯ ^β , ¯ ∇ ^γ  h ^αβ = −h ^λβ R ¯ ^α λγβ + h ^αλ R ¯ λγ

^(2.5.7)

Letus nowbegin with the analysis proper, pointing out onlythe dierent steps of

the al ulation. Boththemetri andthes alareldinthea tion(2.2.17)areexpanded

in aba kground eld and a perturbation

g µν = ¯ g µν + κ h µν

g ^µν = ¯ g ^µν − κ h ^µν + κ ² h ^µ α h ^αν + O(κ ³ )

φ = ¯ φ + κ φ.

^(2.5.8)

Where indi esare raised withthe ba kground metri and geometri quantities ( urva-

ture tensors, ovariantderivatives...) al ulatedwithrespe ttothis metri wearabar.

Totakeintoa ountone-loopee ts itisenoughtoexpand thea tionup toquadrati

order in the perturbations. After expanding, the term linear in the oupling an els

due to the ba kground equationsof motion, namely

∇ ¯ ² φ + 2ΛF ¯ _λ ^′ ( ¯ φ) = 0 R ¯ µν − 1

2 R¯ ¯ g µν − ΛF ^λ ( ¯ φ)¯ g µν − 1

2 ∇ ¯ ^µ φ ¯ ¯ ∇ ^ν φ + ¯ 1

4 g ¯ µν g ¯ ^αβ ∇ ¯ ^α φ ¯ ¯ ∇ ^β φ = 0 ¯

^(2.5.9)

and prime denotes derivative with respe t to

φ

^{. Using the known expansion for the}

s alar urvature the quadrati order operator is

S g = 1 2

Z

d ⁿ x √

¯ g



h ^αβ  1

4 g ¯ αβ g ¯ µν ∇ ¯ ² − 1

4 g ¯ αµ g ¯ βν ∇ ¯ ² + 1

2 g ¯ αµ ∇ ¯ ^β ∇ ¯ ^ν − 1

2 ¯ g µν ∇ ¯ ^α ∇ ¯ ^β + 1

2 g ¯ αβ R ¯ µν − 1

2 g ¯ αµ R ¯ βν − 1

2 R ¯ αµβν + 1

2 g ¯ αµ ∂ β φ∂ ¯ ν φ − ¯ 1

4 g ¯ αβ ∂ µ φ∂ ¯ ν φ ¯

− R + 2ΛF ¯ λ ( ¯ φ) − ¹ 2 g ¯ ^ρσ ∂ ρ φ∂ ¯ σ φ ¯

8 (¯ g αβ ¯ g µν − 2¯g ^αµ g ¯ βν )

! h ^µν

+h ^αβ  1

2 ¯ g αβ ¯ g ^ρσ ∂ ρ φ∂ ¯ σ − ∂ ^α φ∂ ¯ β − Λ¯g ^αβ F _λ ^′ ( ¯ φ)

 φ + φ



− 1

2 ∇ ¯ ² − ΛF λ ^′′ ( ¯ φ)

 φ



(2.5.10)

Atthis stagethe operatorisvery umbersome,but westillhave thefreedom toxthe

gauge in a way that simpliesthe omputation,sin e wehave been areful enough to

in lude the ompensatortoin rease the symmetry to fullDi. Takingthe expresion

χ ν = ¯ ∇ ^µ h µν − 1

2 ∇ ¯ ^ν h − φ∂ ^ν φ ¯

^(2.5.11)

we hoose asgauge xing term

S gf = 1 2

Z

d ⁿ x √

¯ g 1

2ξ ¯ g ^µν χ µ χ ν

^(2.5.12)

whi hafter expanding an beexpressed in the form

S gf = 1 2

Z

d ⁿ x √

¯ g 1

2ξ

 h ^αβ



¯

g µν ∇ ¯ ^α ∇ ¯ ^β − ¯g ^αµ ∇ ¯ ^β ∇ ¯ ^ν − 1

4 g ¯ αβ g ¯ µν ∇ ¯ ²

 h ^µν +2h ^αβ



∂ α φ∂ ¯ β + ¯ ∇ ^α ∇ ¯ ^β φ − ¯ 1

2 g ¯ αβ g ¯ ^ρσ ∂ ρ φ∂ ¯ σ − 1

2 ¯ g αβ g ¯ ^ρσ ∇ ¯ ^ρ ∇ ¯ ^σ φ ¯

 φ

+φ ¯ g ^αβ ∂ α φ∂ ¯ β φ
φ ¯

^(2.5.13)

Letusdenethefollowingtensorwiththedesiredsymmetryproperties,i.e.,symmetri

in

(µν)

^,

(αβ)

^{and under the} inter hange

(µν) ↔ (αβ) C αβµν = 1

4 (¯ g αµ ¯ g βν + ¯ g αν ¯ g βµ − ¯g ^αβ g ¯ µν ) C ^αβµν = ¯ g ^αµ g ¯ ^βν + ¯ g ^αν ¯ g ^βµ − 2

n − 2 g ¯ ^αβ g ¯ ^µν

δ _µν ^αβ = δ ^(α _µ δ _ν ^β)

^(2.5.14)

the full a tion an be writtenas

S g + S gf = 1 2

Z

d ⁿ x √

¯ g 1

2 h ^αβ M αβµν h ^µν + h ^αβ D αβ φ + φE µν h ^µν + φF φ 

(2.5.15)

where the operators are

M αβµν = C αβρσ



−δ µν ^ρσ ∇ ¯ ² + 1 − ξ

ξ g ¯ µν ∇ ¯ ^(ρ ∇ ¯ ^σ) + 2(ξ − 1)

ξ δ _(µ ^(ρ ∇ ¯ ^σ) ∇ ¯ ν) + P _µν ^ρσ



P _µν ^ρσ = −2 ¯ R ^(ρ µ σ)

ν − 2δ (µ ^(ρ R ¯ ^σ) _ν) +



R + 2ΛF ¯ λ ( ¯ φ) − 1

2 g ¯ ^αβ ∂ α φ∂ ¯ β φ ¯



δ ^ρσ _µν + ¯ g ^ρσ R ¯ µν

+ 2

(n − 2) ¯ g µν R ¯ ^ρσ − 1

(n − 2) ¯ g µν g ¯ ^ρσ R + 2δ ¯ _(µ ^(ρ ∂ ν) φ∂ ¯ ^σ) φ − ¯ 1

2 g ¯ µν ∂ ^ρ φ∂ ¯ ^σ φ ¯

− 1

(n − 2) g ¯ ^ρσ ∂ µ φ∂ ¯ ν φ + ¯ 1

2(n − 2) g ¯ µν g ¯ ^ρσ ∂ λ φ∂ ¯ ^λ φ ¯ D αβ = 2(1 − ξ)

ξ C αβρσ ∇ ¯ ^ρ φ ¯ ¯ ∇ ^σ + ξ + 1

ξ C αβρσ ∇ ¯ ^ρ ∇ ¯ ^σ φ − ΛF ¯ λ ^′ ( ¯ φ)¯ g αβ

E µν = 2(ξ − 1)

ξ C µνρσ ∇ ¯ ^ρ φ ¯ ¯ ∇ ^σ + ξ + 1

ξ C µνρσ ∇ ¯ ^ρ ∇ ¯ ^σ φ − ΛF ¯ λ ^′ ( ¯ φ)¯ g µν

F = − ¯ ∇ ² − 2ΛF λ ^′′ ( ¯ φ) + 1

ξ g ¯ ^ρσ ∂ ρ φ∂ ¯ σ φ ¯

^(2.5.16)

in su ha way that interms of the ombined eld

ψ ^A ≡ h ^µν φ

!

(2.5.17)

and in the minimalgauge, orresponding to

ξ = 1

^{, the operator}

S = 1

2 Z

d ⁿ x √

¯ g 1

2 ψ ^A ∆ AB ψ ^B

^(2.5.18)

is minimal,inthe sense that ittakesa Lapla ianform

∆ AB = −g ^AB ∇ ¯ ² + Y AB

^(2.5.19)

with the metri

g AB = C αβµν 0

0 1

!

(2.5.20)

the inverse metri

g ^AB = C ^αβµν 0

0 1

!

(2.5.21)

and the term withoutderivatives

Y AB = C αβρσ P _µν ^ρσ 2C αβρσ ∇ ¯ ^ρ ∇ ¯ ^σ φ − ΛF ¯ λ ^′ ( ¯ φ)¯ g αβ

2C µνρσ ∇ ¯ ^ρ ∇ ¯ ^σ φ − ΛF ¯ λ ^′ ( ¯ φ)¯ g µν −2ΛF λ ^′′ ( ¯ φ) + ¯ g ^ρσ ∂ ρ φ∂ ¯ σ φ ¯

!

(2.5.22)

On the other hand, on e we have an operator in the Lapla ianform (2.5.19), the

one-loop ounterterm (supposing that we work in

n = 4

dimensions) is given by the following oe ientin the heat kernel expansion[11℄

a ₄ = 1 (4π) ^{n 2}

1 360

Z

d ⁿ x √

¯

g tr 180Y ² − 60 ¯ RY + 5 ¯ R ² −

−2 ¯ R µν R ¯ ^µν + 2 ¯ R µνρσ R ¯ ^µνρσ + 30W µν W ^µν

(2.5.23)

and the eld strength is dened through

[ ¯ ∇ ^µ , ¯ ∇ ^ν ]ψ ^A = W _Bµν ^A ψ ^B

^(2.5.24)