Non-Renormalization and Naturalness in a Class of Scalar-Tensor Theories
Claudia de Rham, Gregory Gabadadze, Lavinia Heisenberg, David Pirtskhalava
UUCSD-PTH-12-20NYU-TH-11/11/12
Non-Renormalization and Naturalness in a Class of Scalar-Tensor Theories
Claudia de Rham a , Gregory Gabadadze b , Lavinia Heisenberg a,c andDavid Pirtskhalava d a Department of Physics, Case Western Reserve University, 10900 Euclid Ave,Cleveland, OH 44106, USA b Center for Cosmology and Particle Physics, Department of Physics,New York University, New York, NY, 10003, USA c D´epartement de Physique Th´eorique and Center for Astroparticle Physics,Universit´e de Gen`eve, 24 Quai E. Ansermet, CH-1211, Gen`eve, Switzerland d Department of Physics, University of California, San Diego,La Jolla, CA 92093 USA
Abstract
We study the renormalization of some dimension-4, 7 and 10 operators ina class of nonlinear scalar-tensor theories. These theories are invariant under:(a) linear diffeomorphisms which represent an exact symmetry of the full non-linear action, and (b) global field-space Galilean transformations of the scalarfield. The Lagrangian contains a set of non-topological interaction terms of theabove-mentioned dimensionality, which we show are not renormalized at anyorder in perturbation theory. We also discuss the renormalization of otheroperators, that may be generated by loops and/or receive loop-corrections,and identify the regime in which they are sub-leading with respect to theoperators that do not get renormalized. Interestingly, such scalar-tensor the-ories emerge in a certain high-energy limit of the ghost-free theory of massivegravity. One can use the non-renormalization properties of the high-energylimit to estimate the magnitude of quantum corrections in the full theory. Weshow that the quantum corrections to the three free parameters of the model,one of them being the graviton mass, are strongly suppressed. In particular,we show that having an arbitrarily small graviton mass is technically natural. a r X i v : . [ h e p - t h ] D ec Motivation
Arguably, two of the biggest puzzles of modern cosmology remain the origin ofthe accelerated expansion of the present-day Universe, and the old cosmologicalconstant (CC) problem, arising from a giant mismatch between the theoreticallyexpected magnitude of the vacuum energy, and the tiny value of the observed space-time curvature. Although the resolution of these two puzzles may be related to eachother, General Relativity (GR) fails to address the CC problem, while only beingable to accommodate cosmic acceleration via the postulated dark energy, offeringno insights into its origin.Theories that extend/modify GR at a large distance scale (say at some scale m − ∼ H − , H denoting the present-day value of the Hubble parameter) offer ahope to cancel the vacuum energy by evading S. Weinberg’s no-go theorem, at thesame time describing the accelerated expansion in terms of a small dimensionfulparameter m . Although a satisfactory framework does not yet exist, the resolutionseems to be not too far in the future [1].In such a scenario, a natural question arises as to whether the introduced smallparameter itself (e.g., the graviton mass m ) is subject to strong renormalization byquantum loops, similar to the renormalization of a small cosmological constant .This is one of the questions we will address in the present work.Technically natural tunings are not uncommon within the Standard Model ofparticle physics. According to ’t Hooft’s naturalness argument [2, 3], a physicalparameter c i can remain naturally small at any energy scale E if the limit c i → m e is much smallerthan the electroweak scale for instance, it is technically natural, since quantumcorrections give rise to a renormalization of m e proportional to m e itself, makingthe mass parameter logarithmically sensitive to the UV scale. The reason for thisis simple: taking the m e → λ , on the other hand, there is no symmetry recovered inthe limit λ →
0, and any particle of mass M is expected to contribute to the CC byterms among which are M Λ UV /M , Λ UV /M , with Λ UV denoting the UV cutoff.The smallness of the CC is thus unnatural in the ’t Hooft sense.We will show in the present work, that the introduction of the small gravitonmass m is technically natural, in sharp distinction from the small CC scenario.A Lorentz-invariant modification of GR, such as massive gravity, introduces extrapropagating degrees of freedom, as well as a strong coupling energy scale Λ asso-ciated with these degrees of freedom. Usually, the energy scale Λ is much smallerthan the Planck mass, while significantly exceeding m . For example, on flat space To be clear, by strong renormalization of a parameter we mean an additive renormalizationwhich is proportional to positive powers of the UV cutoff, as opposed to a multiplicative renormal-ization that is only logarithmically sensitive to it; see more on this below. = ( M Pl m ) / ; however it can be much higher on non-trivial backgrounds. Onthe one hand, the presence of strong coupling is typically required to hide unneededextra forces from observations via the Vainshtein mechanism [4, 5] . On the otherhand, the strongly coupled behavior calls for important questions on calculability,quantum consistency, and in this particular case on superluminality, of the massivetheory. The first two are the questions that we will be addressing below. The ques-tion of superluminal propagation is tied to that of potential UV extensions of thesetheories above the scale Λ [6, 7], and will be addressed elsewhere. We will consider a theory of a massless spin-two field h µν , and a scalar π , whichcouple to each other via some dimension 4, 7, and 10 operators; the latter two willbe suppressed by powers of a dimensionful scale Λ . The interactions become strongat the energy scale E ∼ Λ . Nevertheless, we will show that the special structure ofinteractions in this theory guarantees that the operators presented in the tree-levelLagrangian do not get renormalized at any order in perturbation theory.The (non-canonically normalized) Lagrangian of the above-described theory readsas follows [8]: L = − h µν E αβµν h αβ + h µν (cid:88) n =1 a n Λ n − X ( n ) µν (Π) , (1)where E αβµν is the Einstein operator, so that the first term denotes the quadraticEinstein-Hilbert contribution. The dimensionless coefficients a n are tree-level freeparameters (we will fix a = − / X ’s are explicitly given by the following expressions interms of Π µν = ∂ µ ∂ ν π and the Levi-Civita symbol ε µναβ X (1) µν (Π) = ε µαρσ ε ν βρσ Π αβ ,X (2) µν (Π) = ε µαργ ε ν βσγ Π αβ Π ρσ ,X (3) µν (Π) = ε µαργ ε ν βσδ Π αβ Π ρσ Π γδ . (2) X (1 , , are respectively linear, quadratic and cubic in ∂ π , so that the action involvesoperators up to quartic order in the fields.The symmetries of the theory include: (a) linearized diffeomorphisms, h µν → h µν + ∂ ( µ ξ ν ) , which represent an exact symmetry of the full non-linear action ( i.e. including the interactions h µν X ( n ) µν ), and (b) (global) field-space Galilean transfor-mations, π → π + b µ x µ + b . The first of these is a symmetry up to a total derivative. Note however, that a different mechanism of hiding the extra degrees of freedom has beenfound in [1], which does not rely on the Vainshtein effect, but is perturbative. Its virtues will bediscussed later. π , the theory,defined by (1) is ghost-free [8, 9]: it propagates exactly 2 polarizations of the masslesstensor field and exactly one massless scalar; it thus represents a nontrivial exampleof a model with a non-topologically interacting spin-2 and spin-0 fields.We will show in the next section that the operators of this theory remain pro-tected against quantum corrections to all orders in perturbation theory, despitethe existence of non-trivial interactions governed by the scale Λ . Technically, thisnon-renormalization is due to the specific structure of the interaction vertices: theycontain two derivatives per scalar line, all contracted by the epsilon tensors. Then,it is not too difficult to show, as done in the next section, that the loop diagramscannot induce any renormalization of the tree-level terms in (1). Conceptually, thenon-renormalization appears because the tree-level interactions in the Lagrangianare diff invariant up to total derivatives only; on the other hand, the variations ofthe Lagrangian w.r.t. fields in this theory are exactly diff invariant; therefore, noFeynman diagram can generate operators that would not be diff invariant, and theoriginal operators that are diff invariant only up to total derivatives stay unrenor-malized .In a conventional approach that would regard (1) as an effective field theory belowthe scale Λ , there would be new terms induced by quantum loops, in addition tothe non-renormalizable terms already present in (1). Let us consider one-loop termsin the 1PI action. These are produced by an infinite number of one-loop diagramswith external h and/or π lines. The diagrams contain power-divergent terms, thelog-divergent pieces, and finite terms. The power-divergent terms are arbitrary,and cannot be fixed without the knowledge of the UV completion. For instance,dimensional regularization would set these terms to zero. Alternatively, one coulduse any other regularization, but perform subsequent subtraction so that the netresult in the 1PI action is zero.In contrast, the log divergent terms are uniquely determined: they give rise tononzero imaginary parts of various amplitudes, such as the one depicted on Fig.1; thelatter determine the forward scattering cross sections through the optical theorem.Therefore, these pieces would have to be included in the 1PI action.All the induced terms in the 1PI action would appear suppressed by the scale Λ ,since the latter is the only scale in the effective field theory approach (including thescale of the UV cutoff). Moreover, due to the same specific structure of the inter-action vertices that guarantees non-renormalization of (1), the induced terms willhave to have more derivatives per field than the unrenormalized terms. Thereforeat low energies, formally defined by the condition ( ∂/ Λ ) (cid:28)
1, the tree-level termswill dominate over the induced terms with the same number of fields, as well as overthe induced terms with a greater number of fields and derivatives. This propertyclearly separates the unrenormalized terms from the induced ones, and shows thatthe theory (1) is a good effective field theory below the scale Λ . This is similar to non-renormalization of the Galileon operators [10, 11, 12], with diff invariancereplaced by galilean invariance; we thank Kurt Hinterbichler for useful discussions on these points.
4e now move to the discussion of how classical sources enter the above picture.As we will show below, there are similarities to DGP [13] and Galileon [12] theories,but there is also an additional important ingredient that is specific to the presenttheory (1). We will present the discussion for the simplest case a = 0, when theLagrangian can be explicitly diagonalized [8], and will show in the next section thatsetting a = 0 is technically natural. Some of the novel qualitative features readilyapply even when a (cid:54) = 0, however, in this case there are differences too, as we willbriefly discuss below.To this end, it is helpful to perform the field redefinition h µν → h µν + πη µν andrewrite the two nonlinear interactions of the a = 0 theory as follows: a Λ (cid:0) G µν ( h ) ∂ µ π∂ ν π + 3 (cid:3) π ( ∂ µ π ) (cid:1) . (3)The second term in (3) is what appears in DGP (and hence in the cubic Galileontheory). Since in this basis the π field couples to the trace of the stress-tensoras πT /M Pl , the nonlinear term (cid:3) π ( ∂π ) gives rise to the conventional Vainshteinmechanism: for a source of mass M s (cid:29) M Pl , below the Vainshtein radius, r ∗ =( M s /M Pl ) / Λ − , the classical value of the π field is severely suppressed (as comparedto its value in the linear theory) due to the fact that in this regime ∂ π/ Λ (cid:29) .The novelty here is the first term in (3): this term dominates over the second oneboth inside and outside the source (but still inside the Vainshtein radius). Outsidethe source it amounts of having a quartic Galileon in the theory, and since itsphenomenology is well-known [12], we will not discuss it in detail here. However,inside the source we get G µν ( h ) (cid:39) T µν /M Pl , with a good accuracy. Hence, in thisregion the π field gets an additional kinetic term, and the full quadratic term for itcan be written as follows: − ∂ µ π∂ ν π (cid:18) η µν − a T µν Λ M Pl (cid:19) . (4)Thus, for a negative value of a one gets a classical renormalization of the π kineticterm. To appreciate how big this renormalization is we note that the scale in thedenominator of the second term in (4) is M m ; for a graviton mass comparablewith the Hubble parameter H , this is of the order of the critical density of thepresent day Universe. For instance, taking the Earth atmosphere as a source, weget for the kinetic and gradient terms:(1 + | a | )( ∂ π ) − (1 + | a | )( ∂ j π ) . (5)For higher density/pressure sources, such as the Earth itself, or for any Earthlymeasuring device, we get even higher factors of the order 10 and 10 , respectivelyfor the kinetic and gradient terms. The same applies to time-dependent sources, for which, an additional scale due to the timedependence enters the Vainshtein radius [14]. π fluctu-ations above the classical source, δπ = π − π cl , changes qualitatively. Recall that inthe DGP and the standard Galileon theories, the regime of validity of the classicalsolutions can be meaningfully established in the full quantum effective theory dueto the strong classical renormalization of the scalar kinetic terms via the Vainshteinmechanism [11] . Here we get an additional strong classical renormalization of thekinetic term for the fluctuations δπ − ∂ µ δπ∂ ν δπ (cid:0) Z Vµν + | a | Z Tµν (cid:1) , (6)where the first term in (6) is due to the Vainshtein mechanism, which gives rise toa large M s -dependent factor Z V ∼ a ( r ∗ /R Earth ), while the second one is due to theabove-mentioned novel coupling (the first term in (3)).Furthermore, following Ref. [11], the 1PI action can be organized (using somereasonable assumptions about the UV theory) so that the local strong coupling scaledetermining the interactions of the fluctuations δπ schematically reads as follows:Λ eff ( x ) ≡ ( Z V + | a | Z T ) / Λ . Very often Z T (cid:29) Z V , and therefore Z T – although localized in the source – shouldbe taken into account when and if bounds are imposed on the graviton mass from theexistence of this strong scale. For instance, as argued in [12] for the quartic Galileon,the angular part of the quadratic term for the fluctuations is not enhanced by Z V ,presenting a challenge; luckily, the enhancement due to Z T removes this issue in thetheory at hand. Moreover, the Z T -enhancement is present irrespective whether a is chosen to be zero or not – it is solely defined by a nonzero a . Regretfully, thiseffect has not been taken into account in Ref. [15], and the bounds on the gravitonmass obtained there will have to be reconsidered [16].In addition to what we discussed above, there are additional subtleties when a (cid:54) = 0. In this case the Lagrangian (1) cannot be diagonalized by any local fieldredefinition [8]. Hence, the nonlinear mixing term h µν X (3) µν will be present, no matterwhat. Insertions of this vertex into quantum loops will generate higher powersand/or derivatives of the Riemann tensor R µανβ , as well as mixed terms betweenthe Riemann tensor (with or without derivatives) and derivatives of π . In a theorywithout sources all these terms will be suppressed by Λ , again representing a goodeffective field theory below this scale.However, with classical sources included, there should appear a Z -factor supres-sion of the terms containing R µανβ , due to the fact that on nontrivial backgroundsof classical sources there will be a large quadratic mixing between fluctuations of h and π , and the latter has a large kinetic term due to the Z factor as discussedabove. All this will be discussed in Ref. [16]. The right procedure is to first solve for a classical scalar profile in the presence of a source,and then calculate quantum corrections. Of course the opposite order should also give the sameresult once done correctly, however in the latter case one would have to resum quantum correctionsenhanced by large classical terms.
Interestingly, the action given in (1) appears in a certain limit of a recently proposedclass of theories of massive gravity, free of the Boulware-Deser (BD) ghost [18]. Atwo-parameter family of such theories has been proposed in [8, 9]. The theory hasbeen shown to be free of the BD ghost perturbatively in [9], at the full non-linear levelin the Hamiltonian formalism in [19, 20], and covariantly around any backgroundin [21] (see also [22, 23, 24] for a complementary analysis in the St¨uckelberg andhelicity languages, and [25] for a proof in the first order formalism).This class of theories provides a promising framework for tackling the cosmolog-ical constant problem, given that the graviton mass m can be tuned to be aroundthe Hubble scale today. Such a tuning of m with respect to the theoretically ex-pected vacuum energy is of the same order as that of the conventional cosmologicalconstant, m (cid:46) − M ; however, unlike the tuning of the cosmological constant,it is anticipated to be technically natural. The reason for this lies in the fact thatin the m → m → m (cid:54) = 0 deserves a special treatment in thecontext of naturalness .In particular, the theory (1) emerges as the leading part of the ghost-free massivegravity action describing the interactions of the helicity-2 and helicity-0 polarizationsof the graviton in the limit m → , M Pl → ∞ , Λ ≡ ( M Pl m ) / = finite . (7)Beyond this limit, the free parameters of the theory are expected to be renormalized,albeit by an amount that should vanish in the limit (7). As a result, quantumcorrections to the three defining parameters of the full theory (namely the mass m and the two free coefficients a , ) are strongly suppressed. In particular, thegraviton mass receives a correction proportional to itself (with a coefficient thatgoes as δm /m ∼ ( m/M Pl ) / ), thus establishing the technical naturalness of thetheory. Nevertheless, this does not mean that the physical predictions of the theory are discontinuous.As mentioned above, the presence of the Vainshtein mechanism in this model [26, 27, 28, 29], aswell as general [30] extensions of the Fierz-Pauli theory make most of the physical predictionsidentical to that of GR in the massless limit. δm ∝ m property [31]. Besides the fact that the lattertheories are unacceptable, there are two important distinctions between the theorieswith and without BD ghosts. These crucial distinctions can be formulated in the de-coupling limit, which occurs at a much lower energy scale, Λ = ( m M Pl ) / (cid:28) Λ ,if the theory propagates a BD ghost. In the latter case the classical part of thedecoupling limit is not protected by a non-renormalization theorem. As a conse-quence: (a) quantum corrections in ghost-free theories are significantly suppressedwith respect to those in the theories with the BD ghost, and (b) unlike a genericmassive gravity, the non-renormalization guarantees that any relative tuning of theparameters in the ghost-free theories, that is m, a , a , is technically natural . Thelatter property makes any relation between the free coefficients of the theory stableunder quantum corrections .The rest of the paper is organized as follows. We show in Sec. 3 that the interac-tions of the scalar-tensor theory, defined in (1) do not receive quantum corrections toany order in perturbation theory. Identifying the latter theory with the decouplinglimit of massive gravity, we discuss the implications of such Renormalization Group(RG) invariance of relevant parameters in the given limit for the full theory, in Sec. 4showing explicitly that quantum corrections to the graviton mass and the two freeparameters of the potential are significantly suppressed. Finally, we conclude inSec. 5. In this section we present the non-renormalization argument for a class of scalar-tensor theories, defined by the Lagrangian (1). In particular, we will show that thetwo parameters a , do not get renormalized, and that there is no wave functionrenormalization for the spin-2 field h µν .Using the antisymmetric structure of these interactions, we can follow roughlythe same arguments as for Galileon theories to show the RG invariance of these pa-rameters, [10]. The only possible difference may emerge due to the gauge invariance h µν → h µν + ∂ ( µ ξ ν ) , and consequently the necessity of gauge fixing for the tensor field.Working in e.g. the de Donder gauge, the relevant modification of the arguments istrivial: gauge invariance is Abelian, so the corresponding Faddeev-Popov ghosts arefree and do not affect the argument in any way. Moreover, the gauge fixing termchanges the graviton propagator, but as we shall see below, all the arguments thatfollow solely depend on the special structure of vertices and are hence independent For example, a particular ghost-free theory with the decoupling limit, characterized by thevanishing of all interactions in (1) has been studied due to its simplicity ( e.g. see Ref. [32] forone-loop divergences in that model). The non-renormalization of ghost-free massive gravity in thiscase guarantees that such a vanishing of the classical scalar-tensor interactions holds in the fullquantum theory as well. π only appears within interactions/mixings with the spin-2 field in(1). In order to associate a propagator with it, we have to diagonalize the quadraticlagrangian by eliminating the h µν X (1) µν (Π) term. Such a diagonalization gives riseto a kinetic term for π , as well as additional scalar self-interactions of the Galileonform [8], L = − h µν E αβµν h αβ + 32 π (cid:3) π + ( h µν + πη µν ) (cid:88) n =2 a n Λ n − X ( n ) µν (Π) , (8)(here the interactions of the form πX ( n ) (Π) are nothing else but the cubic andquartic Galileons).In the special case when the parameter a vanishes, all scalar-tensor interactionsare redundant and equivalent to pure scalar Galileon self-interactions. This can beseen through the field redefinition (under which the S-matrix is invariant) h µν =˜ h µν + πη µν − a Λ ∂ µ π∂ ν π . We then recover a decoupled spin-2 field, supplemented bythe Galileon theory for the scalar of the form L Gal = − (cid:88) n =0 b n Λ n X ( n ) µν (Π) ∂ µ π∂ ν π , (9)where the Galileon coefficients b n are in one-to-one correspondence with a n and X (0) µν ≡ η µν . The non-renormalization of the theory (1) then directly follows from9he analogous property of the Galileons. For a (cid:54) = 0, such a redefinition is howeverimpossible [8] .We will now show that, similarly to what happens in the pure Galileon theories,any external particle comes along with at least two derivatives acting on it in the 1PIaction, hence establishing the non-renormalization of the operators present in (1). Ofcourse, we keep in mind that these operators are merely the leading piece of the full1PI action, which features an infinite number of additional higher derivative terms.They however are responsible for most of the phenomenology that the theories athand lead to, making the non-renormalization property essential.Consider an arbitrary 1PI diagram, such as the one depicted in Fig. 1. Allvertices in (8) have one field without a derivative, while all the rest come withtwo derivatives acting on them. Any external leg, contracted with a field with twoderivatives in a vertex, obviously contributes to an operator with two derivatives onthe field in the 1PI effective interaction, so if all the external legs were of that kind,this would lead to an operator of the form ∂ j ( ∂ π ) k ( ∂ h µν ) (cid:96) , with j, k, (cid:96) >
0. Theonly possibility of generating an operator with fewer derivatives on some of the fieldscomes from contracting fields without derivatives in vertices with external states.For example, in the interaction V = h µν X (2) µν (Π) ∼ h µν ε αργµ ε βσν γ Π αβ Π ρσ , the spin-2field comes without derivatives, so let us look at an external h µν leg coming out ofthis vertex in an arbitrary 1PI graph, while letting the other two π -particles fromthis vertex run in the loop (of course, all of this reasoning will equivalently applyto any other vertex, such as h µν X (3) µν (Π), or πX (2 , (Π)). Let us denote the external,spin-2 momentum by p µ , while the momenta corresponding to the two π -particlesin the loop are k µ and ( p + k ) µ respectively. The contribution of this vertex to thegraph is given as follows i M ∝ i (cid:90) d k (2 π ) G k G k + p (cid:15) ∗ µν ε αργµ ε βσν γ k α k β ( p + k ) ρ ( p + k ) σ · · · (10)where the Feynman propagator is denoted by G k ≡ ik − i(cid:15) and (cid:15) ∗ µν is the spin-2 polarization tensor, while the ellipses encode information about the rest of thediagram. Now, the key observation is that the term independent of the externalmomentum p , as well as the term linear in it both cancel due to antisymmetricstructure of the vertex. Hence, the only non-vanishing term involves two powers ofthe external spin-2 momentum p ρ p σ i M ∝ i(cid:15) ∗ µν ε αργµ ε βσν γ p ρ p σ (cid:90) d k (2 π ) G k G k + p k α k β · · · , (11)yielding at least two derivatives on the external helicity - 2 mode in the position-space. Thus any external leg coming out of the h µν X (2) µν vertex will necessarily have This can be understood by noting that the h µν X (3) µν coupling encodes information about thelinearized Riemann tensor for h µν , which can not be expressed through π on the basis of thelower-order equations of motion [28]. πX (2) µν vertex (as one can simply substitute h µν by π in theabove discussion).Similarly, if the external leg is contracted with the derivative-free field in vertices h µν X (3) µν and πX (3) µν , their contribution will always involve the external momentum p µ and the loop momenta k µ and k (cid:48) µ with the following structure, i M ∝ i (cid:90) d k d k (cid:48) (2 π ) G k G k (cid:48) G k + k (cid:48) + p f µν ε αργµ ε βσδν k α k β k (cid:48) ρ k (cid:48) σ ( p + k + k (cid:48) ) γ ( p + k + k (cid:48) ) δ · · ·∝ if µν ε αργµ ε βσδν p γ p δ (cid:90) d k d k (cid:48) (2 π ) G k G k (cid:48) G k + k (cid:48) + p k α k β k (cid:48) ρ k (cid:48) σ · · · , (12)where the contraction on the Levi-Civita symbols is performed with either the gravi-ton polarization tensor f µν = (cid:15) ∗ µν or with f µν = η µν depending on whether we aredealing with the vertex h µν X (3) µν or πX (3) µµ . Similar arguments, as can be straight-forwardly checked, lead to the same conclusion regarding the minimal number ofderivatives on external fields for cases in which there are two external states comingout of these vertices (with the other two consequently running in the loops).This completes the proof of the absence of quantum corrections to the two pa-rameters a , , as well as to the spin-2 kinetic term and the scalar-tensor kineticmixing in the theory defined by (1). For massive gravity the action is a functional of the metric g µν ( x ) and four spuriousscalar fields φ a ( x ) , a = 0 , , ,
3; the latter are introduced to give a manifestlydiffeomorphism-invariant description [33, 31]. One defines a covariant tensor H µν asfollows: g µν = ∂ µ φ a ∂ ν φ b η ab + H µν , (13)where η ab = diag( − , , ,
1) is the field space metric. In this formulation, the tensor H µν propagates on Minkowski space. In the unitary gauge all the four scalars φ a ( x )are frozen and equal to the corresponding space-time coordinates, φ a ( x ) = x µ δ aµ andthe tensor H µν coincides with the metric perturbation, H µν = h µν . However, oftenit is helpful to use a non-unitary gauge in which the φ a ( x )’s are allowed to fluctuate.A covariant Lagrangian density for massive gravity can be written as follows, L = M √− g (cid:18) R − m U ( g, H ) (cid:19) , (14)where U includes the mass, and non-derivative interaction terms for H µν and g µν ,while R denotes the scalar curvature associated with the metric g µν .11 necessary condition for the theory to be ghost free in the decoupling limit (DL)is that the potential √− g U ( g, H ) be a total derivative upon the field substitution h µν ≡ g µν − η µν = 0 , φ a = δ aµ x µ − η aµ ∂ µ π [8]. With this substitution, the potentialbecomes a function of Π µν ≡ ∂ µ ∂ ν π and its various contractions with respect to theflat metric η µν . The relevant terms can be constructed straightforwardly by usingthe procedure outlined in Ref. [9].In any dimension there are only a finite number of total derivative combinations,made out of Π, [12]. They are all captured by the recurrence relation [8]: L ( n )der = − n (cid:88) m =1 ( − m ( n − n − m )! [Π m ] L ( n − m )der , (15)with L (0)der = 1 and L (1)der = [Π]. This also guarantees that the sequence terminates, i.e. L ( n )der ≡
0, for any n ≥ L (2)der (Π) = [Π] − [Π ] , (16) L (3)der (Π) = [Π] − ] + 2[Π ] , (17) L (4)der (Π) = [Π] − ][Π] + 8[Π ][Π] + 3[Π ] − ] , (18)where we use the notation: [Π] ≡ TrΠ µν , [Π] ≡ (TrΠ µν ) , while [Π ] ≡ TrΠ µν Π να , withan obvious generalization to terms of higher order in nonlinearity.Then, as argued in [9], the Lagrangian for massive gravity that is automaticallyghost free to all orders in the DL is obtained by replacing the matrix elements Π µν inthe total derivative terms (16)-(18) by the matrix elements K µν , defined as follows: K µν ( g, H ) = δ µν − (cid:112) ∂ µ φ a ∂ ν φ b η ab = (cid:112) δ µν − H µν . (19)Here, and everywhere below, the indices on K should be lowered and raised with g µν and its inverse respectively.This procedure defines the mass term, along with the interaction potential in theLagrangian density of massive gravity [9]: L = M √− g (cid:104) R + m (cid:16) L (2)der ( K ) + α L (3)der ( K ) + α L (4)der ( K ) (cid:17)(cid:105) = M √− g (cid:20) R − m (cid:0) H µν H µν − H + . . . (cid:1)(cid:21) . (20)Since all terms in (15) with n ≥ L ( n )der with n ≥ m, α and α .As is straightforward to see, Minkowski space g µν = η µν with φ a = x a is a vacuumsolution, and the spectrum of the theory (20) contains a graviton of mass m ; thegraviton also has additional nonlinear interactions specified by the action at hand.12he high-energy dynamics of the system is best displayed in the decouplinglimit (DL), defined by (7). Being a direct analog of the nonlinear sigma modeldescription of the high-energy limit of massive spin-1 theories, the decoupling limitof massive gravity features the five polarization states of the graviton, representedby separate helicity states 0 , ± , ±
2. The helicity-2 mode h µν enters linearly inthe decoupling limit, while the helicity-0 mode π is fully non-linear (we will for themoment ignore the vector polarization and will comment on it below). The resultingDL theory takes precisely the form (7). It fully captures the most important featuresof massive gravity, such as the absence of the Boulware-Deser ghost [8] (which hasbeen shown to generalize beyond the DL [21]), the existence of self-accelerating andscreening solutions [1, 26, 34, 35], etc. Moreover, the DL carries all the interactionsthat become relevant within the massless limit and provides a simple illustrationof how the helicity-0 mode decouples from the rest of the gravitational sector for m → The scalar-tensor action given in (1) does not include the DL interactions involvingthe helicity-1 modes A µ of the massive graviton, defined through φ a = δ aµ x µ − η aµ (cid:18) A µ M Pl m + ∂ µ πM Pl m (cid:19) , (21)in (13). So far, their precise form has only been found perturbatively (see for instance[36, 21, 37]). Schematically, to all orders they are given as follows [21] L A = − F + F F (cid:88) n> d n Λ n Π n , (22)where F denotes the field strength for A µ and d n are constant coefficients. Thesevertices can contribute to effective operators involving the helicity-0 mode, π . How-ever, from the explicit form of these interactions it is manifest that every external π , originating from such a vertex will have at least two derivatives on it, in completeanalogy to the case considered above. Taking into account the vector-scalar interac-tions of the form (22) therefore does not change the non-renormalization propertiesof the scalar-tensor part of the action given in (8). Here we denote the canonically normalized fields, obtained by h µν → h µν M Pl and π → πM Pl m bythe same symbols. .2 Implications for the full theory In this subsection we will comment on the implications of the above emergent DLnon-renormalization property for the full theory. Below we will continue to treatmassive gravity as an effective field theory with a cutoff Λ (cid:29) m .We have established previously that in the DL the leading scalar-tensor part ofthe action does not receive quantum corrections in massive gravity: all operatorsgenerated by quantum corrections in the effective action have at least two extraderivatives compared to the leading terms, making the coefficients a i invariant underthe renormalization group flow. This in particular implies the absence of wave-function renormalization for the helicity-2 and helicity-0 fields in the DL. Moreover,the coupling with external matter fields goes as M Pl h µν T µν and thus vanishes as M Pl → ∞ . The non-renormalization theorem is thus unaffected by external quantummatter fields.The DL analysis of the effective action, much like the analogous nonlinear sigmamodels of non-Abelian spin-1 theories [38], provides an important advantage overthe full treatment (see [31] for a discussion of these matters.) In addition to beingsignificantly simpler, the DL explicitly displays the relevant degrees of freedom andtheir (most important) interactions. In fact, as we will see below, we will be able todraw important conclusions regarding the magnitude of quantum corrections to thefull theory based on the DL power counting analysis alone.Now, whatever the renormalization of the specific coefficients α i (and more gen-erally, of any relative coefficient between terms of the form [ H (cid:96) ] · · · [ H (cid:96) n ] in thegraviton potential) is in the full theory (20), it has to vanish in the DL, since α i arein one-to-one correspondence with the unrenormalized DL parameters a i . Let uswork in the unitary gauge, in which H µν = h µν , and for example look at quadraticterms in the graviton potential. We start with an action, the relevant part of which(in terms of the so-far dimensionless h µν ) is L ⊃ − M m (cid:0) (1 + c ) h µν − (1 + c ) h + . . . (cid:1) , (23)where c and c are generated by quantum corrections after integrating out a smallEuclidean shell of momenta and indices are assumed to be contracted with the flatmetric . There is of course no guarantee that the two constants c , are equal, sothey could lead to a detuning of the Fierz-Pauli structure and consequently to aghost below the cutoff, unless sufficiently suppressed. Returning to the St¨uckelbergformalism, in terms of the canonically normalized fields h µν → h µν M Pl , π → πM Pl m (24)the tree-level part ( i.e. the one without c and c ) of the above Lagrangian would We could as well assume that the full non-linear metric contracts indices, since the two casesare indistinguishable at the quadratic level.
L ⊃ − h µν ( ∂ µ ∂ ν π − η µν (cid:3) π ) + . . . . (25)Now, from the DL analysis, we know that this mixing does not get renormalized.What does this imply for the renormalization of the graviton mass and the param-eters of the potential in the full theory?One immediate consequence of such non-renormalization is that in the decouplinglimit, c and c both vanish. To infer the scaling of these parameters with M Pl , letus look at the scalar-tensor interactions that arise beyond the DL. They are of thefollowing schematic form L = (cid:88) n ≥ , (cid:96) ≥ f n,(cid:96) Λ (cid:96) − h ( ∂ π ) (cid:96) (cid:18) hM Pl (cid:19) n , (26) i.e. , they are all suppressed by an integer power of M − compared to vertices arisingin the DL. Then, judging from the structure of these interactions, generically thenon-renormalization theorem for the classical scalar-tensor action should no longerbe expected to hold beyond the DL.This implies that c and c generated by quantum corrections are of the form c , ∼ (cid:18) Λ M Pl (cid:19) k , (27)with k some positive integer k ≥
1, if the loops are to be cut off at the Λ scale (thefact that k needs to be an integer relies on the fact that the theory remains analyticbeyond the DL.) Taking the worst possible case ( i.e. , k = 1), one can directly readoff the magnitude of the coefficients c , , c , ∼ < (cid:18) Λ M Pl (cid:19) ∼ (cid:18) mM Pl (cid:19) / . (28)In terms of the quantum correction to the graviton mass itself, this implies δm ∼ < m (cid:18) mM Pl (cid:19) / , (29)providing an explicit realization of technical naturalness for massive gravity.One can extend these arguments to an arbitrary interaction in the effective po-tential. Consider a generic term of the following schematic form in the unitary gaugeinvolving (cid:96) factors of the (dimensionless) metric perturbation L ⊃ M m √− g (¯ c + c ) h (cid:96) . (30) We are omitting here the part containing the helicity-1 interactions, which can uniquely berestored due to diff invariance of the helicity-2+helicity-1 system, and the U (1) invariance of thehelicity-1+helicity-0 system. In this analysis, the graviton mass m is completely absorbed into Λ , and nothing specialhappens at the scale m as far as the strong coupling is concerned. c denotes the “classical” coefficientof the given term obtained from (20), and c is its quantum correction. Our task isto estimate the magnitude of c based on the non-renormalization of the DL scalar-tensor Lagrangian. Introducing back the St¨uckelberg fields through the replacement h µν → H µν , and recalling the definition of different helicities (21), the quantumcorrection to the given interaction can be schematically written in terms of thevarious canonically normalized helicities as follows (cid:18) hM Pl + . . . (cid:19) (cid:96) (cid:18) hM Pl + ∂AM Pl m + ∂ π Λ + ∂A∂ πM Pl m Λ + ( ∂A ) M m + ( ∂ π ) Λ (cid:19) (cid:96) . The first parentheses denotes a schematic product of √− g and (cid:96) factors of the inversemetric, needed to contract the indices. In the classical ghost-free massive gravity,the pure scalar self-interactions are carefully tuned to collect into total derivatives,projecting out the BD ghost. From the DL arguments, we know that quantumcorrections do produce such operators, e.g. of the form ( ∂ π ) (cid:96) , suppressed by thepowers of Λ . This immediately bounds the magnitude of the coefficient c to be thesame as for the (cid:96) = 2 case c ∼ < (cid:18) Λ M Pl (cid:19) ∼ (cid:18) mM Pl (cid:19) / . (31)Indeed, for c given by (31), we get M m c ∼ Λ and the upper bound on c is thesame as that coming from the mass term renormalization. We have presented a non-renormalization theorem in a special class of scalar-tensortheories, relevant for infrared modifications of gravity.Although these theories feature irrelevant, non-topological interactions of a spin-2 field with a scalar, the couplings corresponding to these interactions do not getrenormalized to any order in perturbation theory. This provides an interestingexample of non-renormalization in non-supersymmetric theories with dimensionfulcouplings.The scalar-tensor theories of this kind arise in the DL of the recently proposedmodels of ghost-free massive gravity. The emergent DL non-renormalization prop-erty, as we have seen, allows one to estimate the magnitude of quantum correctionsto various parameters defining the full theory beyond any limit. In particular, onecan show that setting an arbitrarily small graviton mass is technically natural. Thesignificance of the DL theory is hard to overestimate: it unambiguously determinesall the physical dynamics of the theory at distances Λ − eff ∼ < r ∼ < m − , essentially cap-turing all physics at astrophysical and cosmological scales. Technical naturalness,along with a yet stronger non-renormalization theorem provide perfect predictiv-ity of the theory, enforcing quantum corrections to play essentially no role at these16cales. Moreover, dictated by various physical considerations, one frequently choosesto set certain relations between the two free parameters of the theory, α and α .Non-renormalization in this case means, that such relative tunings of parameters,along with any physical consequences that these tunings may have, are not subjectto destabilization via quantum corrections.In this work, we have not made any assumptions regarding the UV completionof the ghost-free massive gravity. The special structure of the graviton potentialmight lead to a resummation of the infinite number of loop diagrams allowing tostay in the weakly-coupled regime without the need of invoking new dynamics. Theproperties of the DL, including non-renormalization, might be pointing towards sucha simplification of the S-matrix at the apparent strong coupling scale Λ , which mightbecome transparent in a certain alternative field basis [17] (for other proposals forUV behavior, see [39].)We have not addressed the latter questions here and have presented a standardeffective field theory interpretation of massive gravity. Given the effective theoryat the scale Λ , we have shown that the couplings of the leading (decoupling limit)action are RG-invariant as the theory flows towards the infrared. Since it is pre-cisely this part of the action that is responsible for most of the relevant physicsat the astrophysical/cosmological scales, one arrives at rather powerful predictivityproperties of the theory: (a) all the defining parameters of the theory are technicallynatural ; (b) moreover, any choice of relations between them is also technically nat-ural: if one sets a relation at the scale Λ , it remains unchanged at any other lowerscale. This leads to the possibility to study the predictions of the classical theoryat the scales at hand without ever worrying about quantum corrections.We expect similar non-renormalization properties to hold in the recently pro-posed theory of quasi-dilaton massive gravity [40].
Acknowledgments : We would like to thank K. Hinterbichler, R. Rosen andA. J. Tolley for useful discussions. GG is supported by NSF grant PHY-0758032.DP is supported by the U.S. Department of Energy under contract No. DOE-FG03-97ER40546 and would like to thank the Center for Cosmology and Particle Physicsat New York University for hospitality during the final stages of completion of thepaper.
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