Generalized scale invariant theories

Antonio Padilla,1,* David Stefanyszyn,1,† and Minas Tsoukalas2,‡1School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

2Centro de Estudios Científicos, Casilla 1469, Valdivia, Chile(Received 16 December 2013; published 11 March 2014)

We present the most general actions of a single scalar field and two scalar fields coupled to gravity,consistent with second-order field equations in four dimensions, possessing local scale invariance. Weapply two different methods to arrive at our results. One method, Ricci gauging, was known to the literatureand we find this to produce the same result for the case of one scalar field as a more efficient methodpresented here. However, we also find our more efficient method to be much more general when weconsider two scalar fields. Locally scale invariant actions are also presented for theories with more than twoscalar fields coupled to gravity and we explain how one could construct the most general actions for anynumber of scalar fields. Our generalized scale invariant actions have obvious applications to early Universecosmology and include, for example, the Bezrukov-Shaposhnikov action as a subset.

DOI: 10.1103/PhysRevD.89.065009 PACS numbers: 11.25.Hf, 04.50.Kd

I. INTRODUCTION

It could be said that the Universe is nearly scale invariant.This is certainly true of the cosmic microwave backgroundfluctuations recently measured to remarkable accuracy byPlanck [1], as well as the StandardModel of particle physics,whose classical scale invariance is only spoiled by the Higgsmass. One might suspect that this is more than a coincidenceand that scale invariance has some role to play in our searchfor a fundamental theory of nature.Scale invariant theories have been studied in many

different contexts, dating back at least as far as Weyl’sattempts to unify gravity with electromagnetism [2]. It hasrecently been argued by ’t Hooft that they could play animportant role in understanding black hole phenomena andquantum gravity [3], while in early Universe cosmology,there are claims that scale invariance can help us findgeodesically complete solutions [4,5] (see, however, [6]and later [7]). Scale invariance has also been used to identifyuniversality in a wide range of inflationary models [8].However, perhaps the most compelling reason to study scaleinvariance is within the context of nature’s hierarchies. TheStandard Model of particle physics and the concordancemodel in cosmology are both plagued by unnaturally smallmass scales, corresponding to the Higgs mass (∼10−16Mpl)and the cosmological constant (∼10−60Mpl), respectively.In a scale invariant theory there are no mass scales and thehope is that when the symmetry is broken such hierarchiesemerge naturally (see e.g. [9]). Realizing this in practice ischallenging, especially for the latter hierarchy, not leastbecause of the restrictions imposed by Weinberg’s famousno-go theorem. Nevertheless, there have been interesting

proposals.Recently, one of us developed amodel inwhich theStandard Model vacuum energy is sequestered from gravity,exploiting, in part, global scale invariance in the protectedmatter sector [10]. There have also been attempts to use scaleinvariance to stabilize theHiggsmass (seee.g. [11]). Ina seriesof papers, Shaposhnikovand collaboratorshave exploited it topropose a complete cosmological model that attempts toinclude Higgs inflation, a stable Higgs mass, and dynamicaldarkenergy(seee.g. [12]).Forotherusesof scale invariance incosmology and particle physics see, for example, [13–16].There are, of course, many more interesting applications ofscale invariance in the literature, not leastwithin the context ofthe AdS/CFT correspondence [17]. There are even applica-tions within biology [18] and psychology [19].In this paper, we identify a plethora of new (multi)scalar

and (multi)scalar-tensor theories exhibiting scale invari-ance. We consider three separate cases in the followingsections, namely, single scalar theories, biscalar theoriesand finally, theories with more than two scalars. Indeed, ifwe wish to preserve second-order field equations,1 we cansay that the theories presented here exhaust all possibilities,at least in four dimensions for the single scalar and biscalarcases. This is because our starting point is Horndeski’smost general (second-order) scalar-tensor theory [20].Generality is lost for more than two scalars; however alarge class of theories not yet discussed in the literature arepresented by exploiting the multiscalar Horndeski-likeactions presented in [21]. We begin by identifying the

*antonio.padilla@nottingham.ac.uk†ppxds1@nottingham.ac.uk‡minasts@cecs.cl

1Second-order field equations are desirable in order avoidproblems with Ostrogradsky ghosts [22]. If we regard our theoryin the language of effective field theories with a cutoff, however,one could imagine quantum corrections generating higher-orderoperators suppressed by the cutoff scale. These corrections wouldnot introduce any pathologies since the mass of the correspondingghost is up at the cutoff scale [23,24].

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subset of these theories that possess global scale invariance.The path to local scale invariance comes in two forms. Thefirst is to simply identify the subset of (multi-)Horndeskitheories that happens to possess this symmetry. Gravity isknown to play a crucial role in terms of gauging the scaleinvariant ϕ4 theory, so the fact that (multi-)Horndeski is agravitational theory is crucial. The second approach is togauge the globally scale invariant theory directly, using eitherWeylgaugingorRiccigauging[25].The latterwill genericallylead to higher-order field equations unless they fall into thesubset identified in the first approach. Of course, genericallywe do not expect these theories tomaintain scale invariance atthe quantum level without being embedded in some largerconformally invariant theory. Indeed it has recently beenargued that scale invariance plus unitarity requires conformalinvariance in order to be consistent in four dimensions [26].

II. SINGLE SCALAR THEORIES

To illustrate our methods most clearly, we begin bylooking at the case of a single scalar, coupled to gravity. Infour dimensions, the most general such theory with second-order field equations is given by the Horndeski action [20],which we write in the simpler form presented in [27]:

SHorndeski½ϕ; g� ¼Z

d4xffiffiffiffiffiffi−g

p �Kðϕ; XÞ −G3ðϕ; XÞE1

þ G4ðϕ; XÞRþ G4;XE2

þ G5ðϕ; XÞGμν∇μ∇νϕ −G5;X

6E3

�; (1)

where X ¼ − 12ð∇ϕÞ2, En ¼ n!∇½μ1∇μ1ϕ…∇μn�∇μnϕ and

commas denote differentiation i.e. G4;X ¼ ∂G4∂X .We start by demanding global scale invariance such that

under gμν → λ2gμν and ϕ → ϕ=λ, the Horndeski action isinvariant where λ is a constant. For a diffeomorphisminvariant theory, this rescaling of the metric is equivalent toa rescaling of coordinates. Note that we have assumed thatthe scalar has scaling dimension −1. This can be guaran-teed by a simple field redefinition, and given that ourstarting point is the most general theory, with generalpotentials, this is without loss of generality. The resultingaction is nothing more than one would expect fromdimensional analysis with the assumption that no dimen-sionful couplings can appear in the arbitrary functions andwith the scalar field assumed to have a mass dimensionequal to 1. We therefore find that the most general globallyscale invariant subset of the Horndeski action is given by

Sglobal½ϕ;g� ¼Z

d4xffiffiffiffiffiffi−g

p �ϕ4a2ðYÞ−ϕa3ðYÞE1þa4ðYÞϕ2R

þa40ðYÞϕ2

E2þa5ðYÞϕ

Gμν∇μ∇νϕ−a50ðYÞ6ϕ5

E3�;(2)

where we have defined a dimensionless quantityY ¼ X=ϕ4. Note that the familiar − 1

2ð∇ϕÞ2 − μϕ4 theory

is readily obtained by taking a2 ¼ Y − μ. We now considerthe question of local scale invariance, by which wemean invariance under gμν→λ2gμν;ϕ→ϕ=λ, but now λ¼λðxÞ. As stated earlier, there are two possible paths toachieving local scale invariance. The first is to identify thesubset of our globally scale invariant action (2) that alsoexhibits local scale invariance. One could examine thisdirectly and establish conditions on the various functions.However, there exists an argument that allows us to godirectly to the answer and which will generalize nicely inthe multiscalar cases to be studied later. We denote ourlocally scale invariant action by Slocal½ϕ; g�. This actionis unchanged by a scale transformation, gμν → λ2gμν,ϕ → ϕ=λ, and so choosing λðxÞ ¼ ϕðxÞ, we see thatSlocal½ϕ; g� ¼ Slocal½1; ~g�, where ~gμν ¼ ϕ2gμν. To be a subsetof the Horndeski action, we know that Slocal½1; ~g�must havesecond-order field equations, but by Lovelock’s theorem[28] in four dimensions, the most general diffeomorphisminvariant action with second-order field equations built outof ~gμν is

Slocal½1; ~g� ¼Z

d4xffiffiffiffiffiffi−~g

pða ~Rþ bÞ; (3)

where a, b are constants and ~R is the Ricci scalar built outof the metric ~gμν, with a metric connection. Using the factthat ~gμν ¼ ϕ2gμν, we conclude, after some integration byparts, that the unique subset of Horndeski exhibiting localscale invariance is given by the well known action for aconformally coupled scalar field [29]:

Slocal½ϕ;g�¼−12aZ

d4xffiffiffiffiffiffi−g

p �−1

2ð∇ϕÞ2−μϕ4−

1

12ϕ2R

�;

(4)

where μ ¼ b12a. We note that this result is equivalent to the

general action for one scalar presented in [5] where amaximum of two derivatives are allowed in the action. Herewe have proven that terms with greater than two derivativescan play no role if we are to keep second-order fieldequations. It is also worth pointing out that one can alwaysgenerate a scale invariant theory from an action S½~g�, where~gμν ¼ ϕ2gμν. This is done in [30] in order to generate scaleinvariant theories with a single scalar in arbitrary dimen-sions with second-order field equations. In this paper, wehave proven that the theories generated in [30] will be themost general with second-order field equations. Similartechniques were used in [31] to express unimodular gravityas a theory symmetric under transverse diffeomorphismsand Weyl transformations.The second path to local scale invariance is through a

straightforward gauging of the global symmetry. To this

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end we introduce the Weyl vector Wμ transforming asWμ → Wμ þ∇μ log λ and the Weyl covariant derivativeDμ ¼ ∂μ − dWμ acting on an object with scaling dimensiond. We obtain a locally scale invariant action by simplyreplacing all partial derivatives in (2) with the Weylcovariant derivative, or in other words

∂μϕ → Dμϕ ¼ ∂μϕþWμϕ;

∇μ∇νϕ → Dμνϕ ¼ ∇μ∇νϕþ ΩμνðWÞϕ

þ ð4WðμδανÞ − gμνWαÞ�∂αϕþ 1

2Wαϕ

�;

Rμναβ → Rμν

αβ ¼ Rμναβ þ 4δ½α½μΩν�β�; (5)

where ΩμνðWÞ ¼ ∇ðμWνÞ þWμWν − 12gμνW2. Note that

we have dropped terms of the form ∇½μWν� in the above,as such terms are Weyl invariant by themselves. In thissense, our action will be the minimally gauged version of(2) rather than the most general Weyl invariant actioninvolving ϕ, g and W. Our action is, however, still secondorder and it would therefore be natural to add a gaugeinvariant kinetic term for the Weyl vector. In any event, ourgeneralized Weyl action for a single scalar field coupled togravity is given by

Sweyl½ϕ; g;W� ¼Z

d4xffiffiffiffiffiffi−g

p �ϕ4a2ðYÞ − ϕa3ðYÞE1

þ a4ðYÞϕ2Rþ a04ðYÞϕ2

E2

þ a5ðYÞϕ

GμνDμνϕ −a05ðYÞ6ϕ5

E3

�; (6)

where Y ¼ −ðDϕÞ2=2ϕ4, En ¼ n!D½μ1μ1ϕ…Dμn�

μnϕ,

Gμν ¼ Gμν þ 2Ωμν − 2Ωααgμν; R ¼ Rþ 6Ωα

α:

Ricci gauging corresponds to the case where we identify asubset of (6) where the Wμ contributions can be identifiedwith curvature. This is only possible when Wμ only entersthe action through Ωμν, up to a total derivative [25]. Thiscan be guaranteed by treating Wμ and Ωμν as independentfields and ensuring that the variation of Sweyl with respectWμ takes the form

δSweylδWμ

����Ωfixed

¼ ∇νλμν − 2Wνλ

μν þ λννWμ; (7)

where λμν is symmetric in μν. This condition reduces theaction to

Sweyl½ϕ; g;Ω� ¼Z

d4xffiffiffiffiffiffi−g

p �c2ϕ4 −

c32ϕE1 þ c4ϕ2R

�;

(8)

where ci are dimensionless constants and the method ofRicci gauging allows one to tradeΩμν for − 1

2ðRμν − 1

6RgμνÞ

as they transform identically underWeyl transformations. Ingeneral, the metric associated with these curvature termsneed not be the same metric appearing in Sweyl as long as ittransforms the sameway underWeyl transformations, henceintroducing the possibility of generating a locally scaleinvariant bimetric, scalar theory. Initially, let us assume theyare identical in which case it is comforting to note thatRμναβ

reduces to the Weyl tensor and consequently Gμν ¼ R ¼ 0.We find that the resulting Ricci gauged action is

Sweyl½ϕ; g� ¼Z

d4xffiffiffiffiffiffi−g

p �c2ϕ4 þ c3

2ð∇ϕÞ2 þ c3

12ϕ2R

�

(9)

and is equivalent to (4) for appropriate choices of ci,confirming that this action is indeed the most general subsetof Horndeski possessing local scale invariance.As briefly mentioned above, Ricci gauging opens up

the possibility of generating a locally Weyl invariantbimetric, scalar theory built out of two metrics gμν andhμν. Following the same procedure we find that this actiontakes the form

Sweyl½ϕ;g;h� ¼Z

d4xffiffiffiffiffiffi−g

p �c2ϕ4þc3

2ð∇ϕÞ2þc4ϕ2R

þ�6c4−

c32

�ϕ2

�1

12gμνhμν ~R−

1

2gμν ~Rμν

��;

(10)

where ~R and ~Rμν are curvatures associated with hμν.Because of the kinetic mixing between g and h, the action(10) does not fall into the class of bigravity actionspresented in [32], so we cannot be certain that it isghost-free. In fact, in the light of [33], there are stronghints that this theory will indeed contain pathologies.

III. BISCALAR THEORIES

We now examine the case of a biscalar theory coupled togravity and generalize the methods used in the previoussection. Our methods made use of knowledge of the mostgeneral actions built from either N ¼ 0 scalars (EinsteinHilbert) or N ¼ 1 scalars (Horndeski), that do not exhibitthe scale symmetry, to construct the most general action forN ¼ 1 scalars that does possess scale invariance. Therefore,to generalize these arguments and find the most generalscale invariant theory of N scalars coupled to gravity werequire the most general theories without this symmetry for

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either N − 1 or N scalars. Concentrating on the case ofN ¼ 2, let us denote our locally scale invariant actionby Slocal½π;ϕ; g�. This action is unchanged by a scaletransformation gμν → λ2gμν, π → π=λ, ϕ → ϕ=λ and bychoosing λðxÞ ¼ πðxÞ we find that Slocal½π;φ; g� ¼Slocal½1; ~φ; ~g�, where ~ϕ ¼ ϕ=π and ~gμν ¼ π2gμν. As alreadydiscussed, the most general action one can constructfrom ~ϕ and ~gμν is the Horndeski action (1). Thereforethe most general scale invariant theory built from twoscalars and a metric with second-order field equations isgiven by

Slocal½ ~ϕ; ~g� ¼Z

d4xffiffiffiffiffiffi−~g

p �Kð ~ϕ; ~XÞ − G3ð ~ϕ; ~XÞ ~E1

þ G4ð ~ϕ; ~XÞ ~Rþ G4;X~E2

þ G5ð ~ϕ; ~XÞ ~Gμν~∇μ ~∇ν ~ϕ −

G5;X

6~E3

�: (11)

Using the definitions of ~ϕ and ~gμν, we can express thisaction explicitly in terms of ϕ, π and gμν. The relevantterms are

ffiffiffiffiffiffi−~g

p¼ ffiffiffiffiffiffi

−gp

π4;

~R ¼ π−2R − 6π−3□π;

~Gμν ¼ π−4Gμν þ 4π−6∇μπ∇νπ − π−6gμν∇κπ∇κπ − 2π−5∇μ∇νπ þ 2gμνπ−5□π;

~X ¼ π−4Xϕϕ − 2ϕπ−5Xϕπ þ ϕ2π−6Xππ;

~∇μ~∇ν

~ϕ ¼ π−1∇μ∇νϕ − ϕπ−2∇μ∇νπ − 4π−2∇ðμϕ∇νÞπ þ π−2gμν∇απ∇αϕ

þ 4ϕπ−3∇μπ∇νπ − ϕπ−3gμν∇απ∇απ;

~E1 ¼ ~□ ~ϕ ¼ π−3□ϕ − ϕπ−4□π;

~E2 ¼ 2δμ1½μ2δμ3μ4�ðπ−3∇μ1∇μ2φ − φπ−4∇μ1∇μ2π − 2π−4∇μ1φ∇μ2π − 2π−4∇μ2φ∇μ1π þ π−4δμ2μ1∇απ∇αφ

þ 4φπ−5∇μ1π∇μ2π − φπ−5δμ2μ1∇απ∇απÞðπ−3∇μ3∇μ4φ − φπ−4∇μ3∇μ4π − 2π−4∇μ3φ∇μ4π

− 2π−4∇μ4φ∇μ3π þ π−4δμ4μ3∇απ∇αφþ 4φπ−5∇μ3π∇μ4π − φπ−5δμ4μ3∇απ∇απÞ;

~E3 ¼ 6δμ1½μ2δμ3μ4δ

μ5μ6�ðπ−3∇μ1∇μ2φ − φπ−4∇μ1∇μ2π − 2π−4∇μ1φ∇μ2π − 2π−4∇μ2φ∇μ1π þ π−4δμ2μ1∇απ∇αφ

þ 4φπ−5∇μ1π∇μ2π − φπ−5δμ2μ1∇απ∇απÞðπ−3∇μ3∇μ4φ − φπ−4∇μ3∇μ4π − 2π−4∇μ3φ∇μ4π

− 2π−4∇μ4φ∇μ3π þ π−4δμ4μ3∇απ∇αφþ 4φπ−5∇μ3π∇μ4π − φπ−5δμ4μ3∇απ∇απÞ× ðπ−3∇μ5∇μ6φ − φπ−4∇μ5∇μ6π − 2π−4∇μ5φ∇μ6π − 2π−4∇μ6φ∇μ5π þ π−4δμ6μ5∇απ∇αφ

þ 4φπ−5∇μ5π∇μ6π − φπ−5δμ6μ5∇απ∇απÞ;

where Xϕπ ¼ − 12∇ϕ∇π, for example. Note that this action

is invariant under interchange of π and ϕ. This followsautomatically from the freedom to choose the gaugeparameter λ to be either π or ϕ when generating the action.There are many models discussed in the literature which

are a subset of this more general model including the scaleinvariant completions of the Bezrukov-Shaposhnikovactions [12] presented in [5] where one of the scalars istaken to be the Higgs and the Kallosh-Linde inflationarymodel [8] which we recover for 4K ¼ −ð1þ ϕ4=π4 − 2ϕ2=π2Þ, 2G3¼−ϕ=π, 12G4¼1−ϕ2=π2 andG5 ¼ 0.Again, we also compare our general model to the one

presented in [5] and find that constraining (11) to have atmost two derivatives in the action reduces our biscalar

theory to the one presented there. However, in contrast tothe single scalar case discussed above, we have found thatmany more terms can be included, which involve morethan two derivatives in the action, while still keepingsecond-order field equations.Following the previous section, we could also use Weyl

and Ricci gauging and attempt to find this most generalscale invariant action. To use this method, and keepgenerality, we require the corresponding N ¼ 2 theorywithout the symmetry which is, unfortunately, absent fromthe literature. We note that a naive generalization ofHorndeski for N scalars coupled to gravity was presentedin [21]. Although it is proven to give the most general Nscalar action (with second-order field equations) in the

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absence of gravity [34], its covariant analogue was recentlyshown to be missing certain terms [35]. In any event, evenif the most general theory were available, it is clear that theprocess of Ricci gauging the globally scale invariant subsetwould only reproduce (11), a subset, or else include higherequations of motion. One could easily imagine a scenariowhere the latter possibility is realized because the processof Ricci gauging requires one to replace first derivatives ofthe Weyl vector with second derivatives of the metric.

IV. N > 2 SCALAR THEORIES

We now turn our attention to the case of N > 2 scalarfields, coupled to gravity. As mentioned, the most generalcovariant action without the scaling symmetry for N scalarscoupled to gravity is absent from the literature so for N > 2scalars one cannot find the most general action with scaleinvariance. If and when the most general multiscalar tensortheory is identified, let us outline how our first methodshould be applied to that theory.We denote the most general second-order action of N

scalars coupled to gravity as SN ½ϕ1;…ϕN; g� and areinterested in scale invariant theories for which gμν →λ2gμν;ϕi → ϕi=λ, where λ ¼ λðxÞ. Note that we have takeneach scalar to have scaling dimension −1, without loss ofgenerality. Actually, it is more convenient to furtherredefine our scalars, introducing πα ¼ ϕα=ϕN for α ¼ 1…ðN − 1Þ, and πN ¼ ϕN , so that under our scale trans-formation πα are invariant, and πN → πN=λ. We now writeSN ½ϕ1;…ϕN; g� ¼ SN ½π1;…πN; g�.For N scalars, we now let Slocal½ϕ1;…;ϕN; g� denote the

locally scale invariant action. Recall that this is unchangedunder gμν → λ2gμν;ϕi → ϕi=λ, where λ ¼ λðxÞ. We cannow choose λ ¼ ϕN as the transformation parameter andinfer Slocal½ϕ1;…;ϕN; g� ¼ Slocal½π1;…; πN−1; 1; ~g�, wherethe π’s are as defined in the previous paragraph, and~gμν ¼ ϕ2

Ngμν. We have now reduced the action to oneinvolving a metric ~g and only N − 1 scalars; therefore thescale invariant subset could be easily obtained from themost general multi-Horndeski action. We conclude thatthe unique subset of N-scalar Horndeski with local scaleinvariance is given by Slocal½ϕ1;…;ϕN; g� ¼ SN−1½π1;…;πN−1; ~g�, or more explicitly

Slocal½φ1;…;φN; g� ¼ SN−1

�φ1

φN;…;

φN−1

φN;φ2

Ng

�; (12)

where SN−1 has the multi-Horndeski form, but with N − 1,rather than N scalars.Although we do not yet know the form of SN , we do

know of a very large subset [21], so let us conclude thissection by applying the method of Weyl and Ricci gaugingto that theory, which in four dimensions is given by [21]

SN½πl; g� ¼Z

d4xffiffiffiffiffiffi−g

p �AðXij; πlÞ þ AkðXij; πlÞEk

þ ∂B2ðXij; πlÞ∂Xk1k2

Ek1k2þ1

6

∂Bk33 ðXij; πlÞ∂Xk1k2

Ek1k2k3

þ B2ðXij; πlÞR − Bk13 ðXij; πlÞ∇μ∇νπk1Gμν

�;

(13)

where latin indices label the internal index of the field,Xij ¼ − 1

2∇μπi∇μπj for i, j ¼ 1…N, and

Ek1…km ¼ m!∇μ1∇½μ1πk1…∇μm∇μm�πkm :

Recall in passing that the flat space limit of these theories isnow proven to correspond to the most general multiscalarsecond-order theories [34].Following the same methods used for the single scalar

case, we construct the following invariants:

~gμν ¼ π2Ngμν; YNN ¼ XNN

π4N;

YαN ¼ XαN

π3N; Yαβ ¼

Xαβ

π2N;

where α, β ¼ 1…ðN − 1Þ. As expected, global scaleinvariance again corresponds to nothing more than dimen-sional analysis. We now Ricci gauge the global symmetryof (13). As explained in the previous sections, we introducea Weyl vector Wμ and make replacements similar to thosegiven in (5). For the multiscalar case, these are

∂μπα → ∂μπα;

∂μπN → DμπN ¼ ∂μπN þWμπN;

∇μ∇νπα → ∇μ∇νπα þ ½2Wðμ∂νÞ − ðW · ∂Þgμν�πα;∇μ∇νπN → DμνπN ¼ ∇μ∇νπN þ ΩμνðWÞπN

þ ð4WðμδανÞ − gμνWαÞ�∂απN þ 1

2WαπN

�;

Rμναβ → Rμν

αβ ¼ Rμναβ þ 4δ½α½μΩν�β�; (14)

where we recall that πα ¼ ϕα=ϕN , for α ¼ 1;…ðN − 1Þ.Note that even though these are Weyl singlets, we still needto explicitly gauge terms of the ∇μ∇νπα as the metricconnection is not a Weyl singlet. Again we demand that thedependence on Wμ only appears in the combination Ωμν

and then Ricci gauge the remaining action. We find that theRicci gauged version of (13) is given by

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S ¼Z

d4xffiffiffiffiffiffi−g

p �fðYαβ; πγÞπ4N þ hðYαβ; πγÞπN□πN

þ gαðYαβ; πγÞ�πN∇μπα∇μπN þ 1

2π2N□πα

�

−1

6hðYαβ; πγÞπ2NR

�; (15)

where the indices α, β and γ are summed over from 1 toN − 1 and label the index of the fields which do nottransform. This action will, however, only retain second-order field equations when the function h is independent ofYαβ. Clearly for the case of N ¼ 2 the resulting action ismuch less general than (11), highlighting that Riccigauging is unable to produce the most general theories.As expected, this action again reduces to the one presentedin [5] if we ignore terms with greater than two derivatives.Of course, the action (13) does not include a term of the

form [35]

ffiffiffiffiffiffi−g

pδikδjlPμνεη∂μπi∂νπj∂επk∂ηπl (16)

or even more generally

ffiffiffiffiffiffi−g

pFijklðπmÞPμνεη∂μπi∂νπj∂επk∂ηπl; (17)

which nevertheless yields second-order field equations.Here Pμνεη is the double dual of the Riemann tensor andFijkl is symmetric on ik and jl. Given how the metric andthe π’s transform, the globally scale invariant subsets of thisare given by

ffiffiffiffiffiffi−g

pPμνεη½π−4N fαρðπγÞ∂μπα∂νπN∂επρ∂ηπN

þ π−2N fαβρσðπγÞ∂μπα∂νπβ∂επρ∂ηπσ�; (18)

where fαρ and fαβρσ are symmetric in αρ and βσ. One cannow apply the method of Ricci gauging to this, but doing sowill result in a theory that inevitably contains higher-orderfield equations.

V. CONCLUSIONS

In this paper we have presented generalized scaleinvariant theories involving scalar fields coupled to gravity.Indeed, given that one of our starting points has beenHorndeski’s panoptic theory [20], we can say that (4) and(11) are the most general (second-order) theories of thistype admitting local scale invariance for a single scalar and

a biscalar coupled to gravity, respectively. Indeed, our proofconfirms that for a single scalar, the standard conformalcoupling to gravity is the unique theory with local scaleinvariance. When generalized to two scalar fields, however,we see a much wider range of possibilities not previouslydiscussed in the literature. Given the wealth of applicationsof scale invariance, some of which we discussed in theintroduction, these newly identified theories may well openup some exciting new research directions.We have also discussed how one would construct the

most general scale invariant theory for any number of scalarfields as long as the corresponding theory without thesymmetry is known. Without this knowledge, we havefound a very large subclass of these theories using thetechnique of Ricci gauging on the globally scale invariantsubset of N-scalar tensor theories presented in [21]. Evenwithout full generality for more than two scalars, the largesubclass of scale invariant theories that we have now foundopen up new research opportunities especially when con-sidering multiscalar inflationary theories coming from aparent scale invariant theory.Overall we found the technique of Weyl and Ricci

gauging to be less general than the more efficient onepresented at the beginning of each section. This is not reallysurprising given that Ricci gauging requires the Wμ

dependence to take a particular form. Furthermore, it willalso generically generate higher-order field equations, asseen in (15), because Ricci gauging involves replacing firstderivative pieces of the Weyl vector for second derivativesof the metric. We also note that our efficient method is moregeneral than the one presented in [36] which is unable toconstruct the zeroth-order terms appearing in (11). Finally,we remark that it would be interesting to construct andstudy these scale invariant theories within the language oftractor calculus [37].

ACKNOWLEDGMENTS

We thank Paul Saffin, Ricardo Troncoso, ThomasSotiriou, Eleftherios Papantonopoulos and Jorge Zanellifor discussions. M.T. thanks the School of Physics andAstronomy, University of Nottingham for hospitality in thecourse of this work. A. P. was funded by a Royal SocietyURF. D. S. was funded by an STFC studentship. M. T. wasfunded by the FONDECYT Grant No. 3120143. TheCentro de Estudios Cientificos (CECs) is funded by theChilean Government through the Centers of ExcellenceBase Financing Program of Conicyt.

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