Quantum corrections in Galileons from matter loops

Lavinia Heisenberg1,2,*1Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, Ontario, Canada N2L 2Y5

2Départment de Physique Théorique and Center for Astroparticle Physics, Université de Genève,24 Quai Ernest Ansermet, CH-1211 Genève, Switzerland(Received 4 August 2014; published 3 September 2014)

Galileon interactions represent a class of effective field theories that have received much attention sincetheir inception. They can be treated in their own right as scalar field theories with a specific global shift andGalilean symmetry or as a descendant of a more fundamental theory like massive gravity. It is well knownthat the Galileon theories are stable under quantum corrections thanks to the nonrenormalization theoremwhich is not due to the symmetry. We consider different covariant couplings of this Galileon scalar fieldwith the matter field: the conformal coupling, the disformal coupling and the longitudinal coupling. Wecompute the one-loop quantum corrections to the Galileon interactions from the coupling to the externalmatter fields. In all the considered cases of covariant couplings we show that the terms generated byone-loop matter corrections not only renormalize the Galileon interactions but also give rise to higher orderderivative ghost interactions. However, the renormalized version of the Galileon interactions as well as thenew interactions come at a scale suppressed by the original classical coupling scale and hence are harmlesswithin the regime of the effective field theory.

DOI: 10.1103/PhysRevD.90.064005 PACS numbers: 04.50.Kd

I. INTRODUCTION

Cosmology has progressively developed from a philo-sophical to an empirical scientific discipline. Given thehigh precision achieved by the cosmological observations,cosmology is now a suitable arena to test fundamentalphysics. It is witnessing promising attempts to unite thephysics of the large scale structures in the Universe withthe physics of the small scale. Latest observations fromPlanck [1] and BICEP2 [2] with the exquisite precisionhave driven the models of early universe into a corner.Inflation, which describes a phase of accelerated expansionin the very early Universe might explain the initialconditions of the Universe.Observations of the Cosmic Microwave Background,

supernovae Ia, Baryon Acoustic Oscillations and lensinghave led to the cosmological standard model which alsorequires an accelerated expansion of the late Universe,driven by dark energy [3–6]. The physical origin of theaccelerated expansion is still a mystery. In the StandardModel of particles the detection of the missing fundamentalparticle, the Higgs boson, was a revolutionary event. In asimilar way, the missing particles in the standard model ofcosmology, like the graviton, and the resolution to thepuzzle of accelerated expansion and its origin would be asrevolutionary. There are several promising explanatoryattempts to provide an explanation for the acceleratedexpansion of the late Universe.One promising approach corresponds to explaining the

acceleration of the Universe by modifying the geometrical

part of Einstein’s equations. Particularly, weakening gravityon cosmological scales might not only be responsible fora late-time speed-up of the Hubble expansion, but couldalso tackle the cosmological constant problem, whichreflects the large discrepancy between observations andtheoretical predictions. Such promising scenarios arise ininfrared modifications of general relativity like massivegravity or in higher-dimensional frameworks [7–10]. Allthese models of infrared modifications of general relativityare united by the common feature of invoking new degreesof freedom. These degrees of freedom in the consideredspace-time are particles characterized by their masses andspins (or equivalently helicities). These particles are theexcited quanta of the underlying fields. Their Lagrangianare constructed based on the requirement of yieldingsecond order equations of motion, hence with a boundedHamiltonian from below. With these requirements anothersuccessful and interesting class of infrared modificationswas introduced: the Galileons [11]. This Galileon modelrelies strongly on the symmetry of the new scalar degree offreedom, namely the invariance under internal Galilean andshift transformations ϕ → ϕþ bμxμ þ c, and on the ghostabsence. Interestingly Galileon interactions naturally arisein theories of massive gravity which has also been con-structed to be ghost-free [8,10,12]. The Galileon theory hasalso been generalized to the nonflat background case.Direct covariantization might give rise to ghostlike termsin the equation of motion, which has motivated theintroduction of nonminimal coupling between ϕ and thecurvature [13–15]. Even if this covariantization is ghost-free, the Galileon symmetry is lost in curved backgrounds.However, there has been also successful generalizations to*Lavinia.Heisenberg@unige.ch

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the maximally symmetric backgrounds with generalizedGalileon symmetry [16,17]. Interestingly enough, there hasbeen a recent construction of a covariant Galileon theorywithout the requirement of nonminimal interactions andstill avoiding ghost instabilities [18]. Another interestingpoint worth mentioning is the fact that the covariantGalileon interactions also arise from the covariantizationof the decoupling limit of massive gravity [19,20].The Galileon theory exhibits a broad and interesting

phenomenology. However one potentially worrying phe-nomenon is the fact that fluctuations of the Galileon fieldcan propagate superluminally in the regime of interest[11,21,22], i.e. faster than light. Nevertheless, it has beenshown in [23] that closed timelike curves can never arisesince the Galileon inevitably becomes infinitely stronglycoupled and breaks down the effective field theory.Additionally, a dual description to the Galileon interactionshas been discovered by [24–26] in which the originalGalileon theory gets mapped to another Galileon theory bya nontrivial field redefinition ~xμ ¼ xμ þ ∂μϕðxÞ. For agiven very specific Galileon coefficients the Galileoninteractions are dual to a free massless scalar field.Thus, this mapping between a free luminal theory andthe superluminal Galileon theory suggests that the naiveexistence of superluminal propagation can still give rise tocausal theory with analytic and unitary S-matrix [24].These properties together with the presence of a Vainshteinmechanism could be tied to theories which allow for analternative to UV completion such as classicalization.Galileon interactions can be considered as an effective

field theory constructed by the above mentioned restric-tions of symmetry and ghost absence. In order for thetheory to be viable, the Vainshtein mechanism is needed,which on the other hand relies on the presence ofinteractions at an energy scale Λ3 ≪ MPl. From a tradi-tional effective field theory point of view these interactionsare irrelevant operators which renders the theory non-renormalizable, but to contrary to the traditional case,within the Galileon theory these irrelevant operators needto be large in the regime of interest, in the so-called strongcoupled regime ∂2ϕ ∼ Λ3

3. Therefore, one might haveconcerns that the effective field theory could go out ofcontrol in this strong coupling regime where the irrelevantoperators need to be large. Nevertheless, the Galileontheories are not typical effective field theories in the sensethat it is organized in the small parameter expansion of thewhole operator but rather it has to be reorganized in a waythat the derivative now plays the role of the small parameterrather than the whole operator itself [11,27–29]. Thereexists a regime of interest for which ϕ ∼ Λ3, ∂ϕ ∼ Λ2

3 and∂2ϕ ∼ Λ33 even though any further derivative is suppressed

∂3ϕ ≪ Λ43, meaning that the effective field expansion is

reorganized such that the Galileon interactions are therelevant operators with equations of motion with only twoderivatives, while all other interactions with equations of

motion with more than two derivatives are treated asnegligible corrections.Galileon interactions are protected against quantum

corrections via the nonrenormalization theorem. First ofall, the Galileon and shift symmetry will prevent to generatelocal operators by Galileon loop corrections which wouldexplicitly break these symmetries, like potential inter-actions. But this is not enough for the nonrenormalizationtheorem. Quantum corrections might still generate localoperators which are invariant under shift and Galileontransformations, either renormalizing Galileon interactionsthemselves and giving rise to large quantum corrections ofthe strong coupling scale Λ3 or generating operators ofhigher derivative interactions. It is the nonrenormalizationtheorem, which ensures that the Galileon interactionsthemselves are not renormalized at all and that the higherderivative operators are irrelevant corrections in the regimeof validity of the effective field theory. The specific formof the counterterms arising in the 1-loop effective actioncoming from Galileon loops is such that they all come withat least one extra derivative as compared to the originalinteractions. Therefore there is no counterterm which takesthe Galileon form, and the Galileon interactions are hencenot renormalized. This would mean that the Galileoncoupling constants may be technically natural tuned to anyvalue and remain radiatively stable.In this work we address the question of one loop quantum

corrections coming from matter loops. We will consider theGalileon scalar field as a scalar field in its own right, withoutrestricting it to the massive gravity case. The Galileon scalarfield can couple to matter as a conformal mode, ϕT at thelinear level but also as a longitudinal mode, ∂μ∂νϕTμν (eventhough this coupling would vanish for a conserved source),where Tμν is the stress-energy tensor of the external matterfield and T is the trace of it. At the nonlinear level one couldalso consider more generic conformal couplings like ϕ2T orfðϕÞT even though the symmetry would be broken at thelevel of the equations of motion. Naturally, one could alsoconsider derivative couplings of the form ∂μϕ∂νϕTμν

known as disformal coupling (which also arises in massivegravity), or more generally fðϕÞ∂μϕ∂νϕTμν. We will bemainly concentrating on the cases of conformal, disformaland longitudinal couplings.

II. SETUP

In this section we will set up our framework andnotations. For convenience, we will work in Euclideanspace such that the Galileon interactions live on top of a flatEuclidean metric δab. Furthermore, we will use units forwhich h̄ ¼ 1 and we will use the ð−;þ;þ;þÞ signatureconvention. For simplicity, we will assume a massive scalarfield for our matter field which covariantly couples to theGalileon field.Our starting point is the action for the Galileon ϕ and a

massive scalar field χ,

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S ¼Z

d4xðLGal þ LmatterÞ ð2:1Þ

¼Z

d4xLGal þZ

d4x

�1

2ð∂χÞ2 þ 1

2M2χ2

�; ð2:2Þ

where LGal is the Lagrangian for the Galileon interactions

LGal ¼X4n¼0

cnϕUnðΦðxÞÞ ð2:3Þ

with Φμν ¼ ∂μ∂νϕ and the characteristic symmetric poly-nomial invariants

U½Φ� ¼ Eμ1…μ4Eν1…ν4Ynj¼1

Φμjνj

Y4k¼nþ1

ημkνk : ð2:4Þ

In order to have the standard kinetic term for the Galileonscalar field, we can choose c2 ¼ −1=12 and in order tomake the scalar field to have the dimension of mass we canscale the parameters respectively c3 ¼ ~c3=Λ3

3… etc. andreabsorb the dimensionless parameters ~cn. The Galileonfield ϕ can couple to matter χ with an arbitrary coupling. Inthis work we will consider the important representatives ofcommonly used coupling classes. First of all, a natural wayof coupling the Galileon field to the matter is through aconformal coupling of the form ϕT where T ¼ Tμ

μ is thetrace of the associated stress energy tensor of the matterfield χ. As we mentioned, we assume that the fields arepropagating on flat Euclidean space-times and thereforeindices are raised and lowered by the Euclidean metricT ¼ δμνTμν. The stress energy tensor for our matter field issimply given by

Tμν ¼−2ffiffiffiffiffiffi−gp δLmatter

ffiffiffiffiffiffi−gpδgμν

����g¼δ

¼ −∂μχ∂νχ þ1

2δμνðð∂χÞ2 þM2χ2Þ ð2:5Þ

with its trace being T ¼ ðð∂χÞ2 þ 2M2χ2Þ, such that theconformal coupling reads

ϕðð∂χÞ2 þ 2M2χ2ÞMC

ð2:6Þ

We suppressed the interaction by a so far arbitrary scaleMC, which quotes when this interaction becomes impor-tant. In the context of massive gravity, the coupling ofthe Galileon in massive gravity (the helicity-0 degree offreedom of the massive graviton) with the matter fieldcomes from the coupling hμνTμν=MPl, which would meanthat the ϕT coupling is Planck mass suppressedMC ¼ MPl. However, if we consider the Galileon fieldas a scalar field in its own right, then this scale can be

arbitrarily different from the Planck mass. This conformalcoupling can be extended to the nonlinear level ϕ2T at theprize of losing the Galileon symmetry. From the nextorder stress energy tensor one can also construct this typeof nonlinear conformal couplings ϕ2Tμ

μαα where the

tensor Tμναβ arises from the variation of Tμν with respectto the metric,

Tμναβ ¼−2ffiffiffiffiffiffi−gp δ

ffiffiffiffiffiffi−gpTμν

δgαβ

����g¼δ

¼ −∂μχ∂νχδαβ − ∂αχ∂βχδμν

ð2:7Þ

þ 1

2ðδμαδνβ þ δναδμβ þ δμνδαβÞðð∂χÞ2 þM2χ2Þ;

ð2:8Þ

such that the nonlinear conformal coupling wouldbecome

ϕ2ð4ð∂χÞ2 þ 12M2χ2ÞM2

NCð2:9Þ

suppressed by the scale M2NC. Again, in the context of

massive gravity, this coupling would arise fromhμνhαβTμναβ=M2

Pl but here we shall keep the scale arbi-trary. Another important class of possible couplings is thederivative coupling. At the linear level we can also havelongitudinal coupling of the form ∂μ∂νϕTμν which wouldnot contribute in the case of conserved sources, and at thenonlinear level we would respectively have the disformalcoupling of the form ∂μϕ∂νϕTμν. This derivative couplingnaturally arises in the massive gravity and usually comeshand in hand with interesting features. [8,10]. In massivegravity, this interaction would come in at a scale∂μϕ∂νϕTμν=ðMPlΛ3

3Þ, but here we shall consider thiscoupling at an arbitrary scale 1=M4

D.

−∂μϕ∂νϕ∂μχ∂νχ þ 12ð∂ϕÞ2ðð∂χÞ2 þM2χ2Þ

M4D

ð2:10Þ

In the context of Horndeski interactions and screeningmechanisms this disformal coupling has also receivedmuch attention [30–34].For the one loop calculations we will need the Feynman

propagators for the Galileon and the scalar field. TheGalileon field is a massless scalar field and therefore itsFeynman propagator is simply given by

Gϕ ¼ hϕðx1Þϕðx2Þi ¼Z

d4pð2πÞ4

eipμðxμ1−xμ2Þ

p2: ð2:11Þ

On the other hand the matter field we consider here is asimple massive scalar field with the mass M

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Gχ ¼ hχðx1Þχðx2Þi ¼Z

d4kð2πÞ4

eikμðxμ1−xμ

2Þ

k2 þM2: ð2:12Þ

Before starting the computation, let us first remind us ofthe following useful general property due symmetry whencomputing integrals over loops:

1

M4

Zd4kð2πÞ4

k2nka1kb1…kamkbmðk2 þM2Þnþm

¼ 1

2mðmþ 1Þ! δa1b1…ambmJnþm: ð2:13Þ

where Jn stands for the notation

Jn ¼1

M4

Zd4kð2πÞ4

k2n

ðk2 þM2Þn : ð2:14Þ

For our purpose specifically useful cases are mostly

1

M4

Zd4kð2πÞ4

k2ðn−1Þkμkνðk2 þM2Þn ¼

1

4δμνJn ð2:15Þ

1

M4

Zd4kð2πÞ4

k2ðn−2Þkμkνkαkβðk2 þM2Þn ¼ 1

24δμναβJn ð2:16Þ

1

M4

Zd4kð2πÞ4

k2ðn−3Þkμkνkαkβkδkγðk2 þM2Þn ¼ 1

192δμναβδγJn; ð2:17Þ

with

δμναβ ¼ ðδμνδαβ þ δμαδνβ þ δμβδναÞ ð2:18Þ

δμναβδγ ¼ ðδμνδαβδγ þ δμαδνβδγ þ δμβδανδγ

þ δμδδαβνγ þ δμγδαβδνÞ: ð2:19Þ

In this work we will be interested in the running of theinteractions such that we will use dimensional regulariza-tion (or equivalently focus on the log contribution fromcutoff regularization). In this case we have the relation

Jn ¼nðnþ 1Þ

2J1: ð2:20Þ

Another very useful formula that we will be using through-out the paper is the following identity

1

Aα11 Aα2

2 …Aαnn

¼Z

1

0

dx1…dxnδ�X

xi − 1�

×

Qxαi−1i

ðP xiAiÞP

αi

Γðα1 þ � � � αnÞΓðα1Þ � � �ΓðαnÞ

ð2:21Þ

with the Feynman parameters xi.

III. GALILEON LOOPS

The quantum corrections coming from the Galileonself-interactions have already been extensively studied inthe literature and it was successfully shown that they areprotected under quantum corrections [11,27,28,35–37].First of all, the symmetry of the theory, namely the shiftand Galileon symmetry, prevents to generate local oper-ators by loop corrections which breaks explicitly thissymmetry. Nevertheless, this does not forbid the generationof local operators which are invariant under shift andGalileon transformations, either renormalizing Galileoninteractions themselves or generating operators of higherderivative interactions. It is rather the nonrenormalizationtheorem, which ensures that the Galileon interactionsthemselves are not renormalized and that the higherderivative operators are irrelevant corrections in the regimeof validity of the effective field theory ∂nϕ ≪ Λnþ1

3 forn ≥ 3. One can show the nonrenormalization theorem in astraightforward way by realizing that each vertex in anarbitrary Feynman diagram gives rise to interactions with atleast one more derivative.Without loss of generality, let us for a moment concen-

trate on the cubic Galileon interaction ϕEμαρσEνβρσΦμνΦαβ=

Λ33. We can quickly compute the 2-point function contri-

butions coming from this interaction as depicted in Fig. 1.We will let the field with the two derivatives run in the loop.

Mð2pt;3vtÞΛ33

¼ ð−1Þ22!Λ6

3

ð2 · 2ÞZ

d4kð2πÞ4

ð4kαkβq1μq1νðkαkβqμ1qν1 þ kμq1βðkνq1α − 2kαqν1ÞÞÞðk2 þ q21Þ

¼ 2

Λ63

�p81

8ð3J1Þ

�; ð3:1Þ

The counterterm generated by this diagram contains morederivatives per field ð∂4ϕÞ2=Λ6

3 than the classical inter-action □ϕð∂ϕÞ2=Λ3

3. We can generalize this result to anarbitrary Feynman diagram with this interaction at a givenvertex. We can contract an external Galileon field ϕ with

momentum pμ with the Galileon field coming withoutderivatives in this vertex from the cubic Galileon whileletting the other two ϕ-particles run in the loop withmomenta kμ and ðpþ kÞμ. The contribution of this vertexto the scattering amplitude is [28]

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A ∝Z

d4kð2πÞ4GkGkþpEμαρσEνβ

ρσkμkνðpþ kÞαðpþ kÞβ…;

ð3:2Þ

where Gk is the Feynman massless propagator for theGalileon field (2.11). Now, it is a trivial observation that allthe terms which are linear in the external momentumEμαρσEνβ

ρσkμkνkαpβ as well as all the contributions whichare independent of it EμαρσEνβ

ρσkαkβkμkν will cancel owingto the antisymmetric nature of the vertex (carried by theindices in the Levi-Civita symbol). Therefore, the onlynonvanishing term will come in with at least two powers ofthe external Galileon field with momentum pαpβ. This isthe essence of the nonrenormalization theorem of theGalileon interactions. In a similar way, if we contractthe external leg with the derivative free field in vertices ofquartic and quintic Galileon, the same argument straight-forwardly leads to the same conclusion regarding theminimal number of derivatives on external fields.

A ∝Z

d4kð2πÞ4Gk1Gk2 � � � Eμ1…μ4Eν1…ν4

×Yn−1i¼1

kμi kνiðpþ kþ � � �Þμnðpþ kþ � � �Þμn ;Y4

j¼nþ1

ημjνj

ð3:3Þ

There is no counterterm which takes the Galileon form, andthe Galileon interactions are hence not renormalized. Thisguarantees the stability of the Galileon interactions underquantum corrections if one only considers Galileon loops.However, this is not the case when one starts consideringcouplings to other matter fields. In the following section wewill study in great detail the quantum corrections comingfrom matter loops.

IV. MATTER LOOPS

In this section we would like to have a look to thequantum corrections coming from matter one-loops. Wewill compute explicitly the first order loop correctionscoming from the tadpole and two point functions…etc. andtry to generalize the results to the case of n-point functions.In general relativity the helicity-2 degree of freedom cancouple to the matter fields as hμνTμν=MPl and hμνhαβTμναβ=M2

Pl… etc. Now, if we have an additional propagating scalardegree of freedom as the Galileon field, it can also couple

to the matter fields unless we fine-tune the coupling to bezero. The natural way of coupling this additional scalarfield to the matter field is via a conformal couplingϕTμ

μ=MC and ϕ2Tμμαα=M

2NC… etc. If this scalar degree

of freedom couples to ordinary matter, then it can mediate afifth force with a long range of interaction which has neverbeen detected in Solar System gravity tests or laboratoryexperiments. On that account, it is crucial to find ways tohide this extra degree of freedom on small scales. Onecould fine-tune its coupling to matter which is lesssatisfactory. Fortunately, there exist alternatives to fine-tunings thanks to the screening mechanisms that allow us tohide the scalar field on small scales while being unleashedon large scales to produce cosmological effects. Typicalexamples of screening mechanisms are Vainshtein, cha-meleon or symmetron [38–40]. One can basically use themass term, the coupling to matter or the kinetic term of thescalar field to screen its effect. For the Galileon scalar fieldit is the Vainshtein mechanism which is at work. Besidesthese two couplings we will also consider couplings ofthe form 1

M2⋆ϕ2Tμ

μ suppressed with a different scale M⋆. Aswe already mentioned Galileon interactions naturally arisein the decoupling limit of massive gravity with a veryspecific way of coupling to matter. Therefore motivated bymassive gravity, we will also consider disformal couplingsof the form 1

M4D∂μϕ∂νϕTμν between the Galileon field and

the matter fields. Finally we will also consider longitudinalcouplings of the form 1

M3L∂μ∂νϕTμν.

A. Conformal coupling

Our main interest in this work is the quantum correctionscoming from the matter loops. We will consider one-loopcorrections where only matter field runs in the loop. In thissubsection we will first study the quantum correctionscoming from conformal couplings. To start let us have aclose look at the 1-loop contributions to the tadpole and2-point correlation function. The corresponding Feynmandiagrams are represented in Fig. 2. We designate byMð1ptÞ

the 1-loop contribution to the tadpole, by Mð2pt;3vtÞ the1-loop correction to the 2-point correlation function arising

FIG. 1. 1-loop contribution to the 2-point correlation functionfrom the cubic Galileon interaction. Solid lines denote theGalileon scalar field.

FIG. 2. 1-loop contributions to the tadpole and 2-point corre-lation function from the matter coupling. Solid lines denote theGalileon scalar field whereas dashed lines the matter field.

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from the cubic vertex ϕT=MC and by Mð2pt;4vtÞμναβ the 1-loop

correction to the 2-point correlation function arising fromthe quartic vertex ϕ2Tμ

μαα=M

2NC and ϕ2T=M2⋆.

1. Zeroth order in external momenta

To zeroth order in the external momenta we can easilycompute the 1-loop contribution to the tadpole from theconformal coupling ϕT=MC (as represented on the first lineof Fig. 2) yielding

Mð1ptÞp¼0;MC

¼ ð−1ÞMC

Zd4kð2πÞ4

ðk2 þ 2M2Þk2 þM2

¼ 1

2

M4

MCJ1: ð4:1Þ

Now turning to the 2-point correlation function, the 1-loopcontribution arising for the cubic vertex ϕT

MCto zeroth order

in the external momentum is given by

Mð2pt;3vtÞp¼0;MC

¼ ð−1Þ22!M2

Cð2 · 2Þ

Zd4kð2πÞ4

ðk4 þ 4k2M2 þ 4M4Þðk2 þM2Þ2

¼ 2

M2C

�−M4

2ð3J1Þ

�¼ −3

M4

M2CJ1; ð4:2Þ

where the terms in bracket in the first line are combinatoryfactors. Similarly, focusing on the contribution from the

quartic vertex, ϕ2Tμμαα

M2NC

, we obtain

Mð2pt;4vtÞp¼0;MNC

¼ ð−1Þð2ÞM2

NC

Zd4kð2πÞ4

ð4k2 þ 12M2Þk2 þM2

¼ −2

M2NC

ð−4M4J1Þ: ð4:3Þ

Finally, from the nonlinear conformal coupling ϕ2TM2⋆

we have

a second contribution to the two-point function from thequartic vertex

Mð2pt;4vtÞp¼0;M⋆ ¼ ð−1Þð2Þ

M⋆Z

d4kð2πÞ4

ðk2 þ 2M2Þk2 þM2

¼ M4

M2⋆J1: ð4:4Þ

The total 1-loop contribution to the tadpole and 2-pointfunction is thus given by

Mð1ptÞ ¼ 1

2

M4

MCJ1 ð4:5Þ

Mð2ptÞ ¼ Mð2pt;3vtÞ þMð2pt;4vtÞ

¼ M4

�−3M2

Cþ 8

M2NC

þ 1

M2⋆

�J1: ð4:6Þ

This corresponds to the following counterterms at the levelof the action

ŁCT ¼ −ðMð1ptÞðϕÞ þMð2ptÞðϕ2Þ þ � � �Þ ð4:7Þ

¼ −M4J1

�1

2MCðϕÞ þ

�−3M2

Cþ 8

M2NC

þ 1

M2⋆

�ðϕ2Þ þ � � �

�ð4:8Þ

At this point it is not necessary to compute the 3-point orhigher n-point functions explicitly. From the tadpole andthe 2-point function it is already very suggestive that thecoupling ϕT gives rise to appearance of ϕ, ϕ2, ϕ3, ϕ4

terms…etc., potential interactions for ϕ to zeroth order inthe external momenta. (The coupling ϕT breaks sponta-neously the scale invariance.) It is a trivial observation thatthe higher n-point functions will give rise to countertermsof higher potential terms

ŁCT ¼ M4J1Xn

�1

MnCþ 1

MnNC

þ 1

Mn⋆

�ϕn: ð4:9Þ

These quantum contributions are all suppressed by theirown scalesMC,MNC, andM⋆ and hence the scale at which

they become important is when M ≈Mn=4C , M ≈Mn=4

NC , or

M ≈Mn=4⋆ respectively. If the scalar field ϕ is not anarbitrary scalar, but descends from a full-fledged tensorfield hμν as in massive gravity, then all these countertermscoming from one-loop matter quantum corrections arePlanck mass suppressed

ŁMGCT ¼ M4J1

Xn

1

MnPlϕn ð4:10Þ

and so for a Galileon coming from massive gravity one canignore these quantum corrections as long as the mass of theexternal matter field is smaller than Planck mass.

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2. Nonvanishing external momenta

Now we would like also to know what kind ofoperators are generated when we do not put the externalmomentum to zero. In this case we will not only obtainquantum corrections in form of potential interactionsbut also derivative interactions. These generated quan-tum corrections will renormalize the Galileon inter-actions but also give rise to higher derivative operators

with a ghost. From the conformal coupling ϕTMC

there willnot be any contribution with nonzero external momentato the tadpole since the momentum running in the loopis unaware of the external momentum. From the tadpolethere will be only a contribution in the form of apotential linear in the field which we already computedabove. Therefore, the first contribution comes from the2-point function.

Mð2pt;3vtÞp≠0;MC

¼ ð−1Þ22!M2

Cð2 · 2Þ

Zd4kð2πÞ4

ð4M4 þ kμkμð−4M2 þ kνðkν − 2pνÞÞ þ kμpμð4M2 þ kνpνÞÞðk2 þM2Þððp − kÞ2 þM2Þ : ð4:11Þ

In order to perform this integration, we will use the specificrelation of (2.21) with two factors in the denominator

1

Aα1A

β2

¼Z

1

0

dxxα−1ð1 − xÞβ−1

ðxA1 þ ð1 − xÞA2Þαþβ

Γðαþ βÞΓðαÞΓðβÞ : ð4:12Þ

For the above integration we have α ¼ 1, β ¼ 1, A1 ¼k2 þM2 and A2 ¼ ðp − kÞ2 þM2. The key point is now toperform a change of variable such that the mixing of thetwo momentas in ðp − kÞ2 disappears. This is indeedachieved by defining kμ ¼ lμ − ðx − 1Þpμ. In this way,the propagators in the denominator become simply ðl2 þΔ2Þ2 where Δ ¼ M2 þ p2xð1 − xÞ and we can easilyperform the integration. After using the relations (2.15)and performing the x—integration we obtain the followingcontribution

Mð2pt;3vtÞp≠0;MC

¼ 2

M2C

ð3J1Þ8

ð−4M4 − 2M2p2 þ p4ÞÞ: ð4:13Þ

The counterterms arising from the two point functionwould be thus of the form

ŁCT ⊃ −3J14M2

Cð−4M4ðϕÞ2 þ 2M2ð∂ϕÞ2 þ ð□ϕÞ2Þ: ð4:14Þ

The first contribution in the form of a potential correspondsto the contribution we computed above to zeroth order inexternal momentum. Now with a nonvanishing externalmomentum we obtain derivative interactions. The quantumcorrections coming from this 2-point function renormalizethe kinetic term of the Galileon field but also generatesdangerous higher order derivative interactions ð□ϕÞ2. Thisoperator will become important and hence need to beconsidered at a scale ∂2 ∼MC.Similarly as the tadpole, the two point function with the

quartic vertex (on the last line of Fig. 2) coming from theinteractions ϕ2Tμ

μαα

M2NC

and ϕ2TM2⋆

will not have any contributionwith nonzero external momentum since again the internalmomentum in the loop is unaware of the external momen-tum. At the two-point function level the above contribution

will be thus the only one. In order to capture better thegeneral behavior of the n-point function contribution,we shall also perform at this stage the contribution ofthe 3-point function with nonzero external momentum.In Fig. 3 we see the three different contributions to the

3-point function. The first diagram constitutes threevertices with the conformal coupling ϕT=MC giving riseto a contribution of the following form

Mð3pt;3vtÞp≠0 ¼ ð−1Þ3

3!M3C

ð3 · 2Þð4 · 2Þ

×Z

d4kð2πÞ4

Jðk2 þM2Þðq21 þM2Þðq22 þM2Þ

ð4:15Þ

with the momentum conversation relations q1 ¼ ðp1 − kÞand q2 ¼ ðkþ p2Þ and the shortcut notation

J ¼ 8M6 þ 4M4qμ1q2μ

þ kμð2M2q2μð2M2 þ qν1q2νÞþ q1μð4M4 þ q2νð2M2ðkν þ qν1Þ þ kνqα1q2αÞÞÞ

ð4:16Þ

In order to perform this integration involving a denominatorwith three factors we again use Eq. (2.21)

1

A1A2A3

¼Z

1

0

dx1dx2ð3 − 1Þ!

ðx1A1 þ x2A2 þ ð1 − x1 − x2ÞA3Þ3ð4:17Þ

FIG. 3. 1-loop contributions to the 3-point correlation functionfrom the matter coupling.

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Similarly as we did for the 2-point function, we use the trickof completing the square by shifting the integration variablek to kμ ¼ lμ − ðp2μ − p2μx1 − p1μx2 − p2μx2Þ in order toabsorb the terms which are linear in p1 and p2. Thedenominator then again simply becomes ðl2 þ Δ2Þ3 with

this time Δ ¼ M2 − p22ðx1 − 1Þx1 þ ðp1 þ p2Þðp1 þ p2−

2p2x1Þx2 − ðp1 þ p2Þ2x22. We can now perform the inte-

gration over the momentum lμ and the Feynman parameters

x1 and x2. This results in

Mð3pt;3vtÞp≠0 ¼ −ð6J1Þ

6M3C

ð192M4 þ 3p41 þ 4p2

1p22 þ 132p4

2 þ 18M2ðp21 − 11p2

2Þ

þ 2p1 · p2ð3ð−41M2 þ p21 þ 44p2

2Þ þ 67p1 · p2ÞÞ: ð4:18Þ

In an analog way, to the second diagram in Fig. 3 the interactions ϕ2Tμμαα

M2NC

and ϕT=MC will contribute at the respective vertexyielding

Mð3pt;4vt−3vtÞp≠0 ¼ ð−1Þ2

2!M2NCMC

ð3Þð2ÞZ

d4kð2πÞ4

4ð6M4 þ kμðp1μ þ p2μ − kμÞð5M2 þ kνðp1μ þ p2μ − kνÞÞÞðk2 þM2Þððp1 þ p2 − kÞ2 þM2Þ

¼ 3J12M2

NCMCð228M4 þ 147p4

1 − 58p21p

22 þ 3p4

2 þ 48M2ð−10p21 þ p2

2Þ

þ 4p1 · p2ð3ð8M2 − 7p21 þ p2

2Þ þ 7p1 · p2ÞÞ: ð4:19Þ

From the 2- and 3- point function one clearly sees that theone loop contributions are always proportional to M4,M2∂2, or ∂4. The contributions proportional to M4 we canignore here at this stage since they correspond to thecontributions from Eq. (4.21) in the zero external momen-tum limit. On the other hand the contributions with thescaling M2∂2 will be harmless in the sense that they willgive rise to few derivatives per field. The dangerous termsare hence the terms with higher derivatives per field. Thus,the coupling to matter will give rise to counterterms withthe following dangerous higher order derivatives actingon the Galileon field.

ŁCT ¼ −ðMð2ptÞðϕ2Þ þMð3ptÞðϕ3Þ � � �Þ

⊃ M2ð∂ϕÞ2M2

Cþ ð∂2ϕÞ2

M2C

þ ϕð∂2ϕÞ2M3

C

þ□ϕð∂ϕÞ2M2

NCMCþ ϕ

ð∂2ϕÞ2M2

NCMCþ ϕ

ð∂μ∂νϕÞ2M3

C

� � � ð4:20Þ

The quantum corrections coming from the 2-point functionrenormalizes the kinetic term with a scaleM2=M2

C, whereasthe 3-point functions renormalize the cubic Galileoninteraction with a scale 1=M3

C and 1=M2NCMC, respectively.

Therefore, the scale of the interactions need to be chosenabove the scale Λ3 below which the Galileon interactionsare important. For scalesMC ≈ Λ3, the nice structure of theGalileon interaction would be detuned at an unacceptablescale. For the restricted Galileon field coming from massivegravity, the scale at which this ghost would becomeaccessible is close to the Planck scale. The higher n-pointfunctions would give rise to counterterms of the form

ŁCT ¼ J1Xn

�∂2ϕ

MPl

�n

ð4:21Þ

Near sources these quantum corrections are even moresuppressed due to the Vainshtein mechanism ð□ϕÞn−2=ðMn

PlZÞ. In massive gravity this is completely harmlesssince all these quantum corrections will start playing a roleclose to the Planck mass MPl.

B. Disformal coupling

We would like now to draw our attention to the derivativecoupling of the form ∂μϕ∂νϕTμν=M4

D. This coupling has tobe considered in the context of massive gravity. There itarises naturally after shifting the metric perturbations byhμν → ϕημν þ ∂μϕ∂νϕ in order to diagonalize some of theinteractions between the helicity-2 and helicity-0 degreesof freedom. In massive gravity this derivative disformalcoupling brings along important consequences. It plays avery crucial role for the existence of degravitating solutions[10] which is absent in the usual Galileon theories.Furthermore, it plays an important role in the lensingmeasurements since photons can now couple to the scalardegree of freedom [41].We would like now to study the quantum corrections

coming from this derivative coupling. To zeroth order in theexternal momentum this coupling will give zero contribu-tions and hence the potential terms from Eq. (4.21) will notbe generated. Also since already two ϕ fields appear in thecoupling, there will not be any contribution to the tadpoleeither, or to any odd n-point function. Thus, there will beonly contributions to the even number n-point functions.The first contribution will be to the 2-point function tadpoledepicted in the last line of Fig. 2.

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Mð2pt;4vtÞD ¼ ð−1Þ1ð2Þpμpν

M4D

Zd4kð2πÞ4

ð−kμkν þ 12ημνðk2 þM2ÞÞ

k2 þM2¼ M4

4M4Dp2; ð4:22Þ

which would again simply renormalize the kinetic term of the Galileon scalar field ϕ. Next, let us have a look at the nextleading contribution coming from the 4-point function as shown in Fig. 4.

Mð4pt;4vtÞD ¼ ð−1Þ2ð4 · 2 · 2Þp1μp2νp3αp4β

2!M8D

Zd4kð2πÞ4

1

4

Hðk2 þM2ÞðM2 þ ðp1 þ p2 − kÞ2Þ ð4:23Þ

where the shortcut H stands for the following tensor

H ¼ −2kμq1νð−2kαq1β þ gαβðM2 þ k · q1ÞÞ þ gμνð−2kαq1βðM2 þ k · q1Þþ gαβðM4 þ k · q1ð2M2 þ k · q1ÞÞÞ: ð4:24Þ

The energy conservation forces q1 ¼ p1 þ p2 − k and p4 ¼ p1 þ p2 − p3. After performing the integral we obtain the longexpression

Mð4ptÞD ¼ 1

60M8Dð−2p2

1p22ð10M2 þ 3ðp2

1 þ p22ÞÞp3 · p4 − 2p2

1ð10M2 þ p21 − 3p2

2Þðp2 · p3Þp2 · p4

þ pα1ðpβ

2ðð30M4 − 5p41 − 2p2

1p22 − 5p4

2 − 30M2ðp21 þ p2

2ÞÞp3αp4β

þ p3βðð30M4 þ 5p41 þ 18p2

1p22 þ 5p4

2 þ 30M2ðp21 þ p2

2ÞÞp4α − 4ð10M2 þ p21Þp2αp2 · p4ÞÞ

þ 2ðð105M4 − 6p21p

22 þ 20M2ðp2

1 þ p22ÞÞp2αp3 · p4 þ pβ

1ð−p22ð10M2 − 3p2

1 þ p22Þp3αp4β

þ p2αðð−2ð10M2 þ p22Þpγ

1p3β þ ð50M2 þ 3ðp21 þ p2

2ÞÞp2βpγ3 þ 8pγ

1p2βp2 · p3Þp4γ

− 2pγ1p2βð2ðpδ

1 þ 3pδ2Þp3γ − 3p2γpδ

3Þp4δ þ pγ2ð−ð40M2 þ 9ðp2

1 þ p22ÞÞp3βp4γ

þ p3γðð20M2 þ 11ðp21 þ p2

2ÞÞp4β − 4p2βp2 · p4ÞÞÞÞÞÞÞ: ð4:25Þ

Even if the above contribution is a very long expressionwe can still extract useful information. The derivativedisformal coupling gives counterterms of the followingform

ŁCT ¼ −ðMð2ptÞD ðϕ2Þ þMð4ptÞ

D ðϕ4Þ þ � � �Þ

⊃M4

M4Dð∂ϕÞ2 þ M2

M8Dð□ϕÞ2ð∂ϕÞ2 1

M8D□ϕ∂4ϕð∂ϕÞ2

þ � � � ð4:26ÞWe see immediately, that the kinetic term and the quarticGalileon interactions get renormalized. Besides that, thequantum corrections generate counterterms with too many

derivatives acting per field which do not belong to theGalileon class of interactions and contain a ghost degreeof freedom. The kinetic term receives a quantum correc-tion scaling as M4=M4

D which becomes important forM ∼MD. The quartic Galileon on the other handgets renormalized by an operator which scales asM2=M8

D. The higher derivative operator remains negli-gible as long as ∂4=M8

D ≪ Λ43. Again in the case of

massive gravity these contributions would be harmlesssince they would be Planck mass suppressed instead of theMD suppressed.

C. Longitudinal coupling

Even if longitudinal couplings of the form 1M3

L∂μ∂νϕTμν

would vanish for conserved sources and do not originatefrom theories like massive gravity, it is worth studyingquickly the counterterms arising from this type of cou-plings. The scalar field comes already with two derivativesin this longitudinal coupling and we expect that thecounterterms will result in operators with higher derivativesper field. The easiest contribution to compute is the tadpolecontribution

FIG. 4. One-loop contributions to the 4-point function comingfrom the derivative coupling.

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Mð1ptÞ ¼ ð−1ÞM3

Lpμpν

Zd4kð2πÞ4

ð−kμkν þ 12ημνðk2 þM2ÞÞ

k2 þM2

¼ M4

8M3Lp2: ð4:27Þ

Thus, the tadpole already generates a term of the formð□ϕÞM4=M3

L. The contribution to the 2-point function canalso be performed easily

Mð2pt;3vtÞp≠0 ¼ ð−1Þ2

2!M6Lð2 · 2Þpμpνp2αp2β

×Z

d4kð2πÞ4

1

4

Hðk2 þM2Þðq21 þM2Þ ð4:28Þ

where H is given by Eq. (4.24) and where q1 ¼ ðk − pÞand p2 ¼ p because of momentum conversation. Afterperforming the integration this gives

Mð2pt;3vtÞp≠0 ¼ 2

M6Lð3J1Þ

9M4

16p4: ð4:29Þ

The longitudinal coupling gives rise to quantum correctionswith the following counterterms

ŁCT ¼ −ðMð1pt;3vtÞp≠0 ðϕÞ þMð2pt;3vtÞ

p≠0 ðϕ2Þ þ � � �Þ

⊃M4

M3L□ϕþ M4

M6L

ð□ϕÞ2 þ � � � ð4:30Þ

As you can see, this longitudinal coupling naturallygenerates higher derivatives acting per scalar field becom-ing important at a scale close to M ∼ML.

D. Mixed couplings

We can also quickly study an example of Feynmandiagrams in which both the conformal coupling and thederivative couplings contribute at the same time. Thesediagrams mix the scales of the two couplings, therefore thecounterterms of the n-point function will be suppressed bypowers of ðM4

DMCÞn and ðM3LMCÞn, respectively. For our

purpose, it will be enough to study the first contributions ofthese mixed Feynman diagrams. Let us first start with the 3-point function contribution coming from the second dia-gram depicted in Fig. 3 where at the one vertex the couplingϕT=MC and the other vertex the coupling ∂μϕ∂νϕTμν=M4

Dhave to be considered.

Mð3ptÞ ¼ ð−1Þ2ð3 · 2Þð2 · 2Þpμ1p

ν2

2!M4NLMC

Zd4kð2πÞ4

1

2

ð−2kμq1νð2M2 þ k · q1Þ þ gμνð2M4 þ k · q1ð3M2 þ k · q1ÞÞÞðk2 þM2ÞðM2 þ ðp1 þ p2 − kÞ2Þ : ð4:31Þ

We can very easily perform this integral. This diagramgives the following contribution

Mð3ptÞ ¼ 12

M4NLMC

1

24ð−p2

1p22ð6M2 þ p2

1 þ p22Þ

þ p1 · p2ð45M4 − 2p21p

22 þ 6M2ðp2

1 þ p22Þ

þp1 · p2ð18M2 þ p21 þ p2

2 þ 2p1 · p2ÞÞÞ: ð4:32ÞSimilarly, the contribution to the 2-point function with theone vertex being the coupling ∂μ∂νϕTμν=M3

L and the othervertex being the coupling ϕT=MC gives

Mð2ptÞ ¼ ð−1Þ2ð2Þð2 · 2Þ2!M3

LMC

M2

8p2ð15M2 þ 2p2Þ: ð4:33Þ

Summarizing, the Feynman diagrams with the vertices ofmixed couplings to the matter field generate in a verysimilar way contributions as we were obtaining from thepurely conformal or disformal coupling diagrams

ŁCT ¼ −ðMð2ptÞðϕ2Þ þMð3ptÞðϕ3Þ � � �Þ

⊃M4ð∂ϕÞ2M3

LMCþM2ð□ϕÞ2

M3LMC

þM2ð∂ϕÞ2□ϕ

M4DMC

þ ð∂ϕÞ2ð□ϕÞ2M4

DMCþ � � � ð4:34Þ

It is a trivial observation that the contributions to the 3-pointfunction take the form of Galileon interactions as well asnew derivatives interactions not embedded in the Galileoninteractions as we were obtaining above with the differencethat this time the kinetic term scales as M4=ðM3

LMCÞ, thecubic Galileon as M2=ðM4

DMCÞ and the higher orderderivative operators as 1=M4

DMC.

V. STABILITY OF THE COUPLINGS

Another very interesting question is the stability of theclassical couplings. If the couplings themselves receivelarge quantum corrections, it will consequently affect theresults we presented above. In order to illustrate the impactof the quantum corrections of the couplings themselves, letus focus on the conformal coupling ϕT

MC. The question we

want to quickly address is whether or not this couplingreceives large quantum corrections. In order to study thisquestion we will consider two specific diagrams with thecoupling ϕT

MCat the vertices and also the third Galileon

interactions. The for us interesting diagrams are depicted inFig. 5, where in the first diagram the conformal coupling ϕT

MC

acts on the vertices, whereas in the second diagram the thirdGalileon interaction acts on the second vertex. The first

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diagram contributes to hϕχχi through the followingintegration

Mϕχχ1 ¼ ð−1Þ3

3!M3C

ð3Þð4 · 2 · 2Þ

×Z

d4kð2πÞ4

Wðk2 þM2Þðq21 þM2Þðq22Þ

ð5:1Þ

where W stands for

W ¼ 8M6 þ 4M4pμ3q1μ

þ kμð2M2q1μð2M2 þ pν3q1ν

þ p2μð4M4 þ q1νð2M2ðkν þ pν3Þ þ kνpα

3q1αÞÞÞ:ð5:2Þ

The conservation of energy dictates q1μ ¼ p1μ − kμ, q2μ ¼p2μ þ kμ and p3μ ¼ p1μ þ p2μ. Note that now in thepropagator with the momentum q2 the mass does notappear since it corresponds to the ϕ field running in theloop. Therefore, when we use the trick Eq. (4.17) in order towrite the denominator as ðl2 þ Δ2Þ3, an important differ-ence will be also in the Δ

Δ2 ¼ M2ðxþ x2Þ − x22ðp1 þ p2Þ2þ ðp1 þ p2Þðp1 þ p2 − 2p2xÞx2 − p2

2xð−1þ xÞ:ð5:3Þ

We can now perform the integration in l and the Feynmanparameters easily and find

Mϕχχ1 ¼ ð−8Þ

M3C

ð3J1Þ8

ð−16M4 þ 2p42 −M2ð4p2

1 þ 5p22Þ

þ p1 · p2ð−13M2 − p21 þ 4p2

2 þ p1 · p2ÞÞ: ð5:4ÞIn a similar way, we can compute the second diagram inFig. 5. At the two vertices the conformal coupling ϕT=MCacts, whereas on the third vertex the cubic Galileoninteraction ϕEμναβEρσ

αβΦμρΦνσ=Λ33. The contribution of

this diagram is given by

Mϕχχ2 ¼ ð−1Þ3

3!M2CΛ

33

ð5Þð2Þð4 · 2 · 2Þ

×Z

d4kð2πÞ4

W2

ðk2Þðq21Þðq22 þM2Þ ð5:5Þ

with this time

W2 ¼ 2kμðkμð4M4q1νqν1 þ q2νð2M2ðpν2 þ pν

3Þ× q1αqα1 þ pν

2pα3q1βq

β1q2αÞÞ − kνq1μq1ν

× ð4M4 þ q2αð2M2ðpα2 þ pα

3Þ þ pα2p

β3q2βÞÞÞ: ð5:6Þ

Two ϕ-fields are running in the loops, therefore we havetwo massless propagators with momenta kμ and q1μ. Weagain complete the square by shifting the integrationvariable k to kμ ¼ lμ − ðp2μ − p2μx1 − p1μx2 − p2μx2Þand so absorb the linear terms in p1 and p2. Thedenominator again takes the desired form ðl2 þ Δ2Þ3 withthis time Δ ¼ −p2

2ð−1þ x1Þx1 þ ðp1 þ p2Þðp1 þ p2 −2p2x1Þx2 − ðp1 þ p2Þ2x22 −M2ð−1þ x1 þ x2Þ. After per-forming the integration over the momentum lμ and theFeynman parameters x1 and x2 we end up with

Mϕχχ2 ¼ ð−80Þ

3M2CΛ

33

ð3J1Þ16

ðp21ð48M4 þ 6p2

1p22 þ 19p4

2 þ 12M2

× ðp21 þ 4p2

2ÞÞþp1 ·p2ðp21ð48M2 þ 3p2

1 þ 34p22Þ

þ 4p1 ·p2ð3p21 −p2

2 −p1 ·p2ÞÞÞ: ð5:7Þ

From these two diagrams we immediately observe that theconformal coupling itself receives quantum correction interms of the following counterterms

ŁCT ¼ −ðMϕχχ1 ðϕχ2Þ þMϕχχ

2 ðϕχ2Þ þ � � �Þ

⊃M4

M3C

ϕχ2 þ M2

M3C

ϕ∂μχ∂μχ þ 1

M3C

ϕ∂μ∂νχ∂μ∂νχ

þ M4

M2CΛ

33

ϕ∂μχ∂μχ þ M2

M2CΛ

33

ð□ϕÞ∂μχ∂μχ þ � � �

ð5:8Þ

As you can see, the conformal coupling ϕT=MC receivesquantum corrections that scale asM4=M3

C for the mass termand M2=M3

C for the kinetic term, respectively. For massesof the scalar field χ close to M ∼MC these quantumcorrections to the conformal coupling become very impor-tant. The quantum corrections give also rise to couplingswith more derivatives of the form ϕ∂μ∂νχ∂μ∂νχ.

VI. SUMMARY AND DISCUSSION

Historically, Galileon interactions were discovered by theattempt of generalizing the interactions of the decouplinglimit of DGP model [11]. They were constructed by insistingon the symmetry of the helicity-0 mode ϕ of the DGPmodel,namely the invariance under internal Galilean and shifttransformations. From the perspectives of the higher dimen-sional induced gravity braneworld models these symmetriescan be regarded as residuals of the 5-dimensional Poincaré

FIG. 5. One-loop contributions to the 3-point function renorm-alizing the coupling itself.

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invariance. The invariance under these transformationstogether with the postulate of ghost-absence restrict theconstruction of the effective ϕ Lagrangian. One can con-struct only five derivative interactions which fulfill theseconditions as descendants of the Lovelock invariants in thebulk of generalized braneworld models [15–17]. TheGalileon interactions share a very important property: theydo not get renormalized by the quantum corrections arisingfrom the Galileon interactions themselves. It is a commonmisconception in the literature that the nonrenormalizationtheorem is due to the symmetry of the theory. The symmetryguarantees only that there will not be any quantum correc-tions in the form of potential interactions but the symmetrydoes not prohibit the generation of the Galileon interactionsby the quantum corrections. Maybe it would be more fair tosay that the fact that the symmetry is not realized exactly,meaning that the symmetry is fulfilled only up to totalderivatives, plays a crucial role for the nonrenormalizationtheorem.Nevertheless, the nonrenormalization theorem does not

survive when one includes couplings to matter. One cancouple the Galileon scalar field in three different ways tothe matter fields: through linear or nonlinear conformalcoupling, through disformal coupling and finally through alongitudinal coupling. We have considered these threepossible ways of coupling the Galileon to the matter fieldand computed the one-loop quantum corrections to studythe counterterms arising from these couplings. Startingwith the conformal coupling ϕT, we show how the one-loop quantum contributions give rise to potential inter-actions proportional to the mass of the matter field andsuppressed by the scale of the coupling to zeroth order inthe external momenta M4ϕn=Mn

C. Including the contribu-tions with nonvanishing external momentum we observehow the dangerous operators with higher operators per fieldare generated ð□ϕÞ2=M2

C. Another important class ofcouplings is the disformal coupling ∂μϕ∂νϕTμν whicharises naturally in the context of massive gravity.Similarly as in the case of conformal couplings, thequantum corrections from this disformal coupling givesrise to higher derivative operators. The quantum correctionscoming from a longitudinal coupling ∂μ∂νϕTμν share the

same destiny. Furthermore, we studied the interestingquestion whether or not the covariant couplings themselvesare stable under quantum corrections. As an example weperformed the one-loop quantum corrections to the con-formal coupling and found that they get renormalizedsignificantly for scalar masses close to M ∼M3=4

C andM ∼M3=2

C , respectively. Additionally, the appearance ofnew couplings with more derivatives ϕ∂μ∂νχ∂μ∂νχ isunavoidable.Summarizing, in this paper we have shown that quantum

corrections coming from the coupling to the matter fields(i) renormalize the Galileon interactions: the nonre-

normalization theorem protecting the Galileoninteractions does not persist once one considerscouplings to matter field. The standard covariantcouplings give rise to quantum corrections withnonvanishing external momentum which renorm-alize the Galileon interactions.

(ii) generate new higher derivative interactions: in thesame way as the Galileon interactions the couplingto matter fields generates ghostlike interactions withhigher derivative operators.

However, within the regime of the effective field theorythese renormalized Galileons as well as the new higherderivative interactions come in suppressed by the couplingscale. They are harmless as long as(1) the mass of the external scalar field is smaller

than the original classical coupling scale M ≪ MC(2) the derivatives applied on the Galileon field is

smaller than the original classical coupling scale∂ ≪ MC

In the context of massive gravity the original couplingscale is given by the Planck mass and therefore as long asthe mass of the scalar field and the derivatives are smallerthan the Planck mass, the quantum corrections are totallyinsignificant.

ACKNOWLEDGMENTS

We would like to thank Claudia de Rham for usefuldiscussions. This work is supported by the Swiss NationalScience Foundation.

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