The complete gravitational action of the proposed theory consists of the four-dimensional (brane) and the five-dimensional (bulk) parts S=Sbrane+Sbulk,(8)where the two contributions are given respectively in Eqs. (23) and (35) of the next section. Both describe massive gravity (respectively in four and five dimensions), although the corresponding theories differ in nature. The theory on the brane is dRGT massive gravity. The bulk action, on the other hand, describes conventional general relativity, coupled to a pair of conformal scalars with specific boundary conditions that give rise to a gravitational Higgs mechanism on the AdS background of the theory. Both the brane and the bulk theories are discussed in great detail in the next section (respectively in Secs. III A and III B–III C).

In its covariant formulation, the brane (dRGT) theory reduces to general relativity, coupled in a specific way to four diffeomorphism scalars ϕp (p=0…3). In a certain high-energy regime, the fluctuations of these scalars about their expectation values, ⟨ϕp⟩=δμpxμ, describe the extra (helicity-1 and helicity-0) polarizations that a 4D massive graviton propagates in addition to its two general-relativstic (helicity-2) polarizations.

In the parametrically large window of energies m≪E≪Λ3 defining what is known as the decoupling limit of the theory, the most interesting dynamics of massive gravity are captured by an action of the following (schematic) form [8] Sbranedl∼∫braneMPl2h∂2h-MPl2m2[h∂2π+β2h(∂2π)2+β3h(∂2π)3]+hμνTμν,(9)where π denotes the longitudinal scalar polarization of the massive graviton, hμν describes its helicity-2 part and Tμν is the stress tensor of matter. The first term in (9) is the linearized Einstein-Hilbert action, while the next three (those in the parentheses) originate from the mass/potential terms of the graviton. The most general unitary theory of 4D massive gravity is characterized, in addition to the graviton’s mass, by two constant parameters, β2 and β3. Apart from the last term, we have neglected the index structure in (9) (see a more detailed discussion around Eq. (60) below), but we note that it is of a very special type, leading to dynamical equations of at most second order in time, despite the presence of higher-derivative interactions. Moreover, in the given (nontrivial, interacting) limit, the theory is exactly invariant under linearized diffeomorphisms δhμν=∂μξν+∂νξμ,(10)which is due to the fact that the “currents” made out of ∂2π that hμν couples to are kinematically conserved.

As discussed in Sec. I, the longitudinal mode π has no dynamics “on its own”; however, once its quadratic mixing with the helicity-2 polarization hμν is taken into account [the second term in (9)], π acquires a kinetic term of the form MPl2m4π∂2π.(11)This term arises as a result of switching to a new field basis hμν→hμν+πημν, in terms of which the quadratic part of the action is diagonalized. At the same time, the latter field redefinition generates a coupling of π to the trace of the matter stress tensor: LπT=m2πT.(12)Canonically normalizing the fields, hμν→MPl-1hμν and π→(MPlm2)-1π, and carefully inspecting the action reveals that the interactions of π become strong at the low scale Λ3, while the coupling of π to matter is order-one in units of MPl-1, leading to a “fifth force” of gravitational strength. To avoid conflict with observations, one needs to rely on precisely the strongly coupled dynamics of π, which suppresses the contribution of this field to the classical Newtonian potential [3,5]. Nevertheless, Λ3 is the true quantum cutoff of the theory and predictivity of massive gravity at characteristic distance scales below Λ3-1∼103  km relies on making certain assumptions about the putative short-distance completion of the dynamics [10].

We wish to argue that these issues are remedied upon inclusion of the bulk part of the action [16]. While the bulk theory is described in detail in the next section, we note that at energies and momenta lower than L-1, it effectively reduces to the 5D Fierz-Pauli theory of a free AdS graviton h¯ab with mass m¯≪L-1. The (linearly) diffeomorphism-invariant action of this theory, including the bulk Stückelberg vector Va, reads Sbulk≃M53∫d5x-g¯[-14∇ch¯ab∇ch¯ab+12∇ch¯ab∇bh¯ac-12∇ah¯∇bh¯ab+14∇ah¯∇ah¯-2L2(h¯abh¯ab-12h¯2)-m¯24((h¯ab-2∇(aVb))2-(h¯-2∇·V)2)],(13)where ∇ denotes the covariant derivative with respect to the background metric g¯ab. Carefully accounting for the nontrivial commutation rules of AdS covariant derivatives reveals that Va is a massive vector with mass mV2=8L-2, in agreement with Eq. (4) of the introductory section.

From now on, we will concentrate on the helicity-2 (h¯ab) and the helicity-0 (Va∼∇aΠ) components of the bulk graviton, and zoom onto distance scales much smaller than the graviton’s Compton wavelength, m¯→0, which defines the decoupling limit of the bulk theory: Sbulkdl∼∫bulkM53∇h¯∇h¯-M53m¯2L-2(∇Π)2.(14)The first term in this expression schematically denotes the 5D Einstein-Hilbert term, expanded to the quadratic order on anti–de Sitter space, while the second is the kinetic term for Π, which arises thanks to the vector Va being massive on AdS. As discussed above, the latter kinetic term would be absent on flat space, and the dynamics of Π would only arise through mixing with h¯; in the present case of AdS background, however, this mixing is completely negligible. The theory (14) is exactly invariant under the linearized bulk diffeomorphisms δh¯ab=∇aξ¯b+∇bξ¯a,(15)however, as we will be exclusively working in the gauge corresponding to an unbent brane at z=0, we have to require ξ¯z(x,0)=0. It is in this gauge that we identify the boundary fields entering the brane theory (9) with their bulk counterparts: hμν(x)=δμaδνbh¯ab(x,0)≡h¯μν|,π(x)=Π|.(16)From now on, a vertical stroke will denote evaluation on the brane. Keeping the brane at z=0, one can further fix the gauge so that h¯zz=h¯μz=0 is true in the bulk. The residual freedom then corresponds to choosing the bulk gauge parameters as ξ¯μ(x,z)=L2(z+L)-2ωμ(x),ξ¯z(x,z)=0, which generates the four-dimensional brane diffeomorphisms (10), corresponding to ξμ(x)=ωμ(x) [16].

Let us summarize our setup. The simplified model that we wish to explore in the rest of this section is specified by the total (bulk+brane) decoupling limit action Sdl=∫braneMPl2h∂2h-MPl2m2[h∂2π+β2h(∂2π)2+β3h(∂2π)3]+hμνTμν+∫bulkM53∇h¯∇h¯-M53m¯2L-2(∇Π)2,(17)supplemented by the identification (16) of the brane and the bulk fields. The bulk theory is understood to be gauge-fixed so that the brane is unbent at z=0 (that is, the “brane bending mode” is gauged away). The action is then invariant under the linearized bulk diffeomorphisms (15) with ξ¯z|=0, and the corresponding 4D reparametrizations of the brane (10). One can use this freedom to impose h¯zz=h¯μz=0 in the bulk, which still leaves residual gauge invariance, under which the bulk and the brane fields transform with {ξ¯μ(x,z)=L2(z+L)-2ωμ(x),ξ¯z=0} and ξμ(x)=ωμ(x).

The dynamical equations for h¯ab and Π that follow from varying the bulk action (17) describe respectively a massless spin-2 field (fully analogous to the graviton of general relativity) and a massless scalar on anti–de Sitter space. It is well known that the massless tensor field gets localized on a positive tension brane in AdS [19], and so does a massless scalar [20]. That is, even if one does not include the “bare” action Sbrane to start with, their brane images hμν(x) and π(x) acquire 4D kinetic terms, thereby mediating four-dimensional interactions between brane sources.

With the given normalization of modes, the induced 4D kinetic terms are of the following form [19,20]⁴

There is also an induced kinetic mixing between hμν and π [16], but this will not be important in the following discussion.

The small mixing with the helicity-2 polarization does still give rise to a coupling of π^ to the matter stress tensor of the form LπT=αMPlπ^T,(20)with α∼mL. Unlike purely 4D massive gravity, however, this coupling is tiny as the graviton’s mass is parametrically smaller than the AdS curvature scale. This renders the fifth forces, mediated by the helicity-0 graviton very small, and in fact, there is no vDVZ discontinuity [1,2] even on flat space—the graviton mass can be taken to zero and with it, the coupling of the longitudinal mode to a matter stress-tensor goes to zero as well.

The enhanced kinetic term of the helicity-0 graviton results in the weak coupling of the theory all the way down to distances well below any macroscopic scale. To show this, one can examine the most relevant interactions of π^ which, as it turns out, are given by the following terms Sdl∼∫braneπ^∂2π^-β2M⋆3εεh^(∂2π^)2-β3Λ⋆6εεh^(∂2π^)3+⋯,(21)where the suppression scales have been defined as M⋆3∼MPlL2≫1L3,Λ⋆6∼MPl2mL3.(22)The second of these scales, Λ⋆, is lower than M⋆, although it is still much higher than Λ3. Indeed, for a GUT-scale AdS curvature and a Hubble-scale graviton mass, we have Λ⋆∼107  GeV, to be compared with Λ3∼10-22  GeV: the cutoff of the brane theory has increased by 29 orders of magnitude.

Furthermore, setting β3=0—a technically natural choice in the theory at hand [21]—results in further increase in the strong coupling (energy) scale. In this case, the scale suppressing the strongest interaction is M⋆. This scale is greater than AdS curvature, however, and it turns out not to have physical meaning—the true cutoff of the 4D theory is set by L-1. This is because, as we discuss in more detail in the next section, the latter scale marks the point where new bulk states—the “radial Higgs” modes involved in the 5D gravitational Higgs mechanism—start to become “visible” to the 4D brane observer.

For the most general choice of the parameters at hand (that is, without assuming m∼m¯ and M53L∼MPl2), the expressions for the two scales M⋆ and Λ⋆ generalize to those given in Eq. (64) of the next section.

We have stated above that it is impossible to modify the Nambu-Goldstone sector of massive gravity within a unitary and local theory, and yet we have argued that a consistent modification exists which, at least in some limit, gives rise to a local kinetic term for π. How is that possible? The resolution to the apparent paradox lies in the fact that strictly speaking, all kinetic terms, induced from the bulk are nonlocal—they can be thought of as arising from integrating out a continuum of particles, which includes an infinite number of states, lighter than Λ3. Nevertheless, the couplings of the continuum states to external fields depend on their 4D mass; in particular, light states couple very weakly to the brane fields, making it possible to make sense of “integrating them out,” which results in local physics at energies below L-1. This is analogous to how effective 4D physics emerges on the Randall-Sundrum II (RS II) brane [19].

Let us first specify the brane action. This is given by dRGT massive gravity—the unique unitary and local nonlinear extension of the Fierz-Pauli theory [8,9]. In its diffeomorphism-invariant formulation, in addition to the 4D metric γμν, the theory features 4 auxiliary scalar fields ϕp, with their flavor index running over p=0,…,3. Explicitly, the action reads Sbrane=M422∫d4x-γ[R4-m24∑n=24αnUn(K)-2Λ4],(23)where R4 and Λ4 are the 4D Ricci scalar and cosmological constant respectively, αn are constant parameters, m is the graviton’s mass (which fixes α2=2 in four dimensions) and M4 is the “bare” 4D Planck mass. The full effective 4D Planck scale that we will refer to as MPl in this section, will receive an extra contribution from bulk dynamics. Furthermore, the mass/potential terms Un can be written with the help of the 4D totally antisymmetric symbol ε as follows⁵

The term linear in Kνμ (corresponding to n=1) leads to a tadpole on the Minkowski background and thus obstructs having a 4D Poincaré-invariant vacuum. We will discard it in the rest of this paper.

The dynamical equations that follow from varying the action (23) admit a flat-space solution with the following expectation values ⟨γμν⟩=ημν,⟨ϕp⟩=δμpxμ.(26)On this background, the scalars’ internal indices mix with the spacetime ones, and we will sometimes not make distinction between the two. One can use diffeomorphism invariance of the theory to fix unitary gauge, in which the four scalars are frozen to their background values, ϕμ=xμ. In this gauge, (23) describes a Lorentz-invariant theory of the metric alone.

Away from unitary gauge and at sufficiently high energies, the most interesting dynamics of massive gravity feature the helicity-2 (hμν) and helicity-0 (π) polarizations, defined respectively by the following equations γμν=ημν+hμν,ϕp=δμp(xμ+ημνvν)vμ=aμ-∂μπ(27)(the helicity-1 mode, on the other hand, is captured by the Lorentz vector aμ). The high-energy limit of interest is then defined as a double scaling limit MPl→∞,m→0,Λ3=finite,(28)in which the relevant part of the action (excluding the helicity-1 mode) becomes Ldl=-M424hμν(E^h)μν-M42m24[εεh∂2π+β2εεh(∂2π)2+β3εεh(∂2π)3]+hμνTμν.(29)Here we have defined β2=(3α3+4)/4 and β3=(α3+4α4)/4, and used the simplified notation, given in Eq. (24).

As remarked multiple times above, π has no “independent” dynamics: it only receives its kinetic term through mixing with the helicity-2 polarization of the massive graviton. This kinetic term is “small” in the sense discussed in the previous section, leading to the low strong coupling scale of the theory, as well as an order-one coupling of π to matter (in gravitational units). To avoid these problems, it would be tempting to try giving π an “independent” kinetic term by means of modifying the action for the Nambu-Goldstone sector ϕp of the theory. However, one can show that any such (local) modification would clash with unitarity: the dynamics of the Nambu-Goldstone sector of massive gravity is uniquely determined by locality and the absence of extra, pathological d.o.f.

One way out could be to give up locality. A generic nonlocal modification of the theory would put it on shaky grounds, as it would likely reintroduce problems with the basic principles of quantum field theory such as unitarity, causality, etc. Nevertheless, we know of at least one way to make a 4D field theory nonlocal without spoiling consistency: embed it in a certain local higher-dimensional spacetime.

We would like the higher-dimensional theory to describe AdS gravity in Higgs phase, as outlined in the introductory section. In order to better understand how the Higgs phenomenon works in this case, we will start with discussing the representation theory of the global symmetry group of 5D AdS: SO(2,4).

The irreducible, unitary representations of SO(2,4) are labeled by the eigenvalues of the maximal compact subgroup SO(2)⊗SO(4)≅SO(2)⊗SU(2)+⊗SU(2)-, denoted respectively by E, s+ and s-. In particular, E measures energy of a particle in units of AdS curvature L-1, which is different from the particle’s Lagrangian mass m¯. The precise relation between the two is (see e.g., [23] and references therein) m¯2L2=E(E-4)s=0,m¯2L2=(E+s-2)(E-s-2),s≥1,where we have denoted the spin of the particle in question by s. An integer spin-s state with energy E forms an irreducible representation that we will refer to as D(E,s/2,s/2). Moreover, unitarity requires that energy be bounded from below: E≥s+2, and this inequality is only saturated for massless particles. In terms of the SO(2,4) representation theory, the Higgs mechanism can be understood as the following statement: in the massless limit E→s+2, a spin-s representation D(E,s/2,s/2) becomes reducilble, splitting into the following direct sum (see Refs. [14,15,23–25] and references therein): D(E,s/2,s/2)→E→s+2D(s+2,s/2,s/2)⊕D(s+3,(s-1)/2,(s-1)/2).(30)The first representation on the right-hand side is the massless spin s particle, which “eats up” a massive NG boson of spin s-1 (the second term in the direct sum) and becomes massive. For the case of a spin-2 graviton in 5D, the NG boson is a vector of energy E=5, corresponding to the Lagrangian mass mV2L2=8. The action that describes such a particle is precisely the one in Eq. (4), discussed in the context of the Stückelberg formulation of the theory in the introductory section.

The above discussion has only concerned the linearized limit of massive gravity. Can the gravitational Higgs mechanism be embedded into a full-fledged nonlinear theory on anti de Sitter space? To answer this question, we need to first understand the origin of the NG vector D(5,1/2,1/2) in such a nonlinear theory. In the scenario of interest, D(5,1/2,1/2) will arise as part of a two-particle Hilbert space, formed by a direct product of spin-0 representations, H2=D(E+,0,0)⊗D(E-,0,0) [14]. For concreteness, we will consider the case that the two representations in the product both stem from a conformally coupled AdS scalar. Such scalars can be quantized in two different ways on anti de Sitter space (depending on the specific boundary conditions one imposes at the spacetime boundary), corresponding to E±=(d±1)/2 (so that E++E-=d) [26]. In the case of AdS4, the explicit expression for H2 can be found e.g., in Ref. [14], and it is straightforward to generalize the formula to AdS5 [25]: D(E+,0,0)⊗D(E-,0,0)=∑n=0∞∑s=0∞D(E++E-+s+2n,s/2,s/2).(31)One can see that the NG vector D(5,1/2,1/2) does indeed appear in the two-particle Hilbert space of scalar representations of SO(2,4) in d+1=5. Importantly, E+ and E- necessarily have to be different for the mechanism to work: had we chosen the same scalar representations on the left-hand side of (31), D(5,1/2,1/2) would be eliminated from the two-particle Hilbert space by Bose statistics [14]. Denoting the two scalars with energy E+ and E- respectively by ϕ1 and ϕ2 and taking into account parity of the composite NG vector V with respect to the interchange ϕ1↔ϕ2, we have the following relation between V and the two constituent scalars [14] V=ϕ1∇ϕ2-ϕ2∇ϕ1,(32)Moreover, the CFT dual of V is expressed in terms of the CFT duals O1,2 of ϕ1,2 as OV=O1∂O2-O2∂O1.(33)The NG vector has precisely the right SO(2,4) quantum numbers to mix with the graviton and become “eaten up” in the Higgs phase of the theory. Moreover, D(5,1/2,1/2) is the only such state: it is easy to convince oneself that no other particle in H2 can have a linear mixing with the graviton. Indeed, AdS5 particles can mix at the level of the quadratic action only if their SO(2,4) quadratic Casimirs C2 are identical.⁶

The quadratic Casimir enters the dynamical equation for a spin-s field on AdS in the following way: (Δ-C2)Ψa1…as=0, where Δ is the Lichnerowicz operator which commutes with covariant derivatives and traces and reduces to -□ in the flat-space limit of the theory.

As a digression, we note that a similar mechanism of mass generation for the electroweak gauge bosons would operate in the Standard Model of particle physics, had its vacuum been AdS. The reason is that, as it turns out, any chiral gauge symmetry (such as the SU(2)L of the SM) is bound to be broken down to a vector subgroup by the AdS-invariant boundary conditions of fermions, leading to mass generation for the W-boson even in the absence of the Higgs condensate [27]. In this case, the required NG bosons are provided by the geometric bound states of chiral SM fermions, much in the same way as a bound state of conformal scalars gives rise to the gravitational Higgs mechanism.

The above-described Higgs mechanism for gravity involves mixing between one-particle (the massless AdS graviton) and two-particle (the composite vector) states, and therefore the graviton mass will only arise at the 1-loop level. In order to compute it, one can look at the correction to the graviton self-energy from couplings to the two conformal scalars D(E±,0,0).

In fact, it will prove more interesting to start with a setup involving two spin-2 states on AdS5, with different “Planck masses” M1 and M2, each coupled to a free conformal scalar. (In the end we will go back to the case with one spin-2 particle, taking the limit in which the second decouples.) This corresponds to having a theory of “bigravity,” defined by the following action [28] Sbulk=∑i=1,2∫dd+1x-gi[Mid-12(Ri-2Λ)-12giab∂aϕi∂bϕi-d-18dϕi2Ri]-d-E+2∫ddx-γ1ϕ12+∫ddx-γ2(E-2ϕ22+ϕ2n2a∂aϕ2).(35)Here γ1,2 are the induced metrics on the brane and nia are the unit (outward) normals, satisfying giabnianib=1. So far this is a theory consisting of two decoupled sectors, and as such it is invariant under two distinct sets of diffeomorphisms, diff1 and diff2, corresponding to transforming the two pairs of bulk fields (metric plus scalar) separately. For a negative cosmological constant Λ, the dynamical equations of the theory admit a solution with ⟨ϕi⟩=0, both metrics describing AdS space of curvature L2=-d(d-1)/2Λ (in order to have a flat boundary, one needs to add compensating tension terms for each of the two metrics at z=0 as well, which we leave implicit here).

Solving the dynamical equation for a conformally coupled scalar on AdS yields the following behavior near the brane (see Appendix A) ϕ(z→0,x)≈(L+z)E+β(x)+(L+z)E-(α(x)-LE+-E-β(x))=(E+-E-)LE+-1zβ(x)+LE-α(x)+O(z2/L2).(36)Such a scalar can be quantized with two AdS-invariant (Dirichlet or Neumann) boundary conditions, corresponding to {β=0,α≠0} or {α=0,β≠0}. These quantization rules give rise to the SO(2,4) representations D(E+,0,0) and D(E-,0,0) respectively [26,29], and we have added the boundary terms in (35) to enforce just these boundary conditions, with the identification E1=E+ and E2=E- [30]. Consider now adding a small perturbation to the boundary action: Sλ=-λ(E+-E-)∫ddx-γ1(E-ϕ1+n1a∂aϕ1)ϕ2.(37)This enforces mixed boundary conditions on ϕ1(ϕ2) in (35), corresponding to mostly describing the state with AdS energy E+(E-), but with a small admixture of the state with energy E-(E+). More explicitly, the new boundary term (37) correlates the boundary conditions between the two scalars so that their behavior near the brane becomes ϕ1(z→0,x)≈(E+-E-)LE+-1zβ1(x)+LE-α1(x),ϕ2(z→0,x)≈LE-β2(x)+(E+-E-)LE+-1zα2(x),(38)where the α-coefficient of one scalar is related to the β-coefficient of the other one as follows α1,2=∓λ(E+-E-)β2,1.(39)The motivation for choosing such boundary conditions comes from the gauge/gravity duality, and the small coupling λ will have a simple interpretation in terms of the dual CFT description of the theory, as will be discussed in more detail in the next section.⁷

There is in fact an ambiguity in (37) as to which induced metric one should couple the perturbation to: γ1, γ2, or both. At the 1-loop level we will be interested in, this will not matter—all that matters is the background value of the metric (which is the same for both γ1 and γ2). Beyond the 1-loop level, we choose to couple the perturbation term to γ1, as this is the field whose “Planck mass” we will eventually send to infinity, effectively decoupling its fluctuations.

To proceed, we note that a simple rotation brings us back to the basis of fields with independent boundary conditions χ1=11+λ˜2(ϕ1+λ˜ϕ2),(40)χ2=11+λ˜2(-λ˜ϕ1+ϕ2),(41)where we have defined λ˜≡λ(E+-E-). Moreover, the fields χ1,2 have the right boundary conditions to describe irreducible scalar representations D(E+,0,0) and D(E-,0,0) and can therefore be quantized in the AdS-invariant way. Notice, however, that the AdS-invariant quantization will necessarily break one combination of two diffeomorphisms we started with. This is because χ1 and χ2, being linear combinations of ϕ1 and ϕ2, do not have well-defined transformation properties under the full group diff1⊗diff2, but only under the diagonal combination of the two diffs, which remains unbroken. This, as we will see, results in mass generation for one combination of the original spin-2 particles.

This mass, denoted below by m¯, has been computed previously by several authors in the AdS/CFT limit, which formally corresponds to moving the brane all the way to the AdS boundary z′=0; the calculation for a single bulk spin-2 state has been done in Refs. [14,31], and has been generalized to the case with two spin-2 particles in [28,32]. In the two Appendixes, we extend these calculations in our “regularized AdS/CFT” setup with the brane located at z′=L in the Poincaré patch coordinates (7). Our calculation is significantly more involved, but we find that the expression for m¯ is not corrected at the leading order in L, compared to the results obtained in the AdS/CFT limit.

To proceed, we note that in terms of the quantized χ-fields the matrix of 2-point functions is diagonal: ⟨χiχj⟩=δijG˜Ei,(42)where we have defined E1,2=E±, and G˜E1,2 are the scalar AdS propagators, satisfying the appropriate (Dirichlet and Neumann) boundary conditions. The explicit expressions for these propagators are given in Appendix A. Furthermore, as discussed extensively around Eq. (31), the Nambu-Goldstone vector D(5,1/2,1/2) that the graviton needs to eat up to become massive is only contained in a tensor product of different scalar SO(2,4) representations. This means, in particular, that nonzero contributions to the 1-loop graviton mass come exclusively from the operators in (35) that involve both fields χ1 and χ2. These operators, as can be inferred from Eqs. (35), (40), and (41), all couple to the same combination (h1-h2)/2 of the original spin-2 particles, which will therefore acquire mass at one loop. The resulting quadratic spin-2 Lagrangian schematically reads [33] L=14(M13+M23)h(0)E^h(0)+M13M23M13+M23h(m)E^h(m)-σh(m)h(m),(43)where we have defined h(0)=2M13M23M13+M23(1M23h1+1M13h2),(44)h(m)=12(h1-h2).(45)The last term in (43) describes the loop-generated Fierz-Pauli mass with σ=λ2/32πL5, while the first two terms denote the kinetic terms for the massless and the massive combinations of the original fields. The action (43) also makes it clear that the massless and the massive spin-2 fields couple with different strengths (i.e., have different Planck masses). In particular, in the limit M1→∞ the massless spin-2 state decouples from all external fields, while the massive one still has a finite “Newton’s constant” of order M2-3 [33]. The decoupled massless spin-2 mostly corresponds to the original field h1, while the massive combination—which we will sometimes refer to as the “graviton” below—is mostly h2, and its mass is of order m¯2=σM23=λ232πM23L5.(46)We will be mostly interested in this “single-graviton” limit in the rest of this paper.

To close the present discussion, we remark that one can integrate out the two scalars ϕ1 and ϕ2 from the bulk action (35), thereby arriving at (two copies of) general relativity, corrected by the Coleman-Weinberg “potential” that depends on the two metrics alone: Sbulk(g1,g2)=∑i=1,2∫dd+1x-gi[Mid-12(Ri-2Λ)]+i2∫ddxlogdetΔ(x,z;x′,z′).(47)The matrix Δ, defined as Δ(x,z;x′,z′)=δ2S/δϕi(x,z)δϕj(x′,z′), can be formally expressed in terms of the following differential operator on anti de Sitter space Δ=[-g1(□1↔-d-14dR1)δ1iδ1j+-g2(□2↔-d-14dR2)δ2iδ2j]δd+1(x,z;x′,z′)-[-γ1(E-+n1·∇1↔)δ1iδ1j--γ2(E-+n2·∇2↔)δ2iδ2j]δ(z)δ(z′)δd(x,x′)--γ1λ(E+-E-)(E-(δ1iδ2j+δ1jδ2i)+n1·∇1δ1iδ2j+n1·∇1′δ1jδ2i)δ(z)δ(z′)δd(x,x′),where ∇i′ (∇i) is the covariant derivative with respect to the metric gi, acting on (un)primed coordinates, ni·∇i↔≡12(ni·∇i+ni·∇i′), and similarly for the covariant Laplacian □↔.

The expression for the determinant becomes slightly simpler in the limit M1→∞, in which the fluctuations of the tensor g1 decouple. In this case, this field can be substituted by its background value g¯.

Having specified both the boundary and the bulk gravitational actions, Eqs. (23) and (35), we are finally in a position to study the strong coupling phenomenon in the effective 4D theory of massive gravity on the brane. We will work in the limit of the single massive spin-2 state in the bulk (with the other, massless spin-2 state decoupled, as discussed at the end of the previous subsection). This state will be referred to as h¯ab (while the full bulk metric is gab=g¯ab+h¯ab). Moreover, its coupling—the higher-dimensional Planck mass, denoted before by M2—will be renamed into M5. As we have discussed in the previous subsection, the mass of the bulk graviton, m¯∼λ˜2/M53L5, is parametrically smaller than the two other scales in the problem, M5 and L-1, which we will assume are only mildly separated from each other: L-1≲M5. Apart from h¯ab, the 5D AdS theory features a tower of “bound states” of the two scalars ϕ1,2, that have various spins and Lagrangian mass parameters of order L-1 and larger. Because the underlying “fundamental” theory (35) is weakly coupled at energies and momenta below the 5D Planck scale, the “effective” theory of the massive h¯ab plus the bound states is as well.

A short comment on the precise meaning of “the low-energy limit” of the bulk theory is in order. Such a limit is in fact somewhat subtle on AdS, since at energies/momenta, lower than L-1 (which we are assuming is a rather high energy scale in this work), the effects of the background curvature become order-one important. Perhaps a more intuitive definition of the low-energy regime—which we adopt throughout in this paper—arises from the perspective of a 4D brane observer. Indeed, such an observer lives on flat space, and can therefore probe the gravitational interactions by conventional means, e.g., by scattering 4D matter particles and measuring the amplitudes. As a matter of fact, even from the 4D perspective there is a subtlety, as the gravitational sector of the effective brane theory is strictly speaking nonlocal, containing, in addition to 4D gravity, a gapless continuum of (Kaluza-Klein) states. Nevertheless, one can still make sense of the 4D low-energy effective field theory: the extra bulk states are “invisible” to an observer, confined to the brane and working at energies/momenta lower than AdS curvature L-1 [19]. We will return to this point below.

Back to the bulk theory. At the level of the quadratic action, its relevant part is given by the Fierz-Pauli theory of Eq. (13), where the (composite) Stückelberg vector Va, defined in terms of the constituent scalars in Eq. (32), transforms under the linearized 5D diffeomorphisms as δVa=ξ¯a(x,z).(48)At the same time, the helicity-2 field h¯ab transforms as δh¯ab=∇aξ¯b+∇bξ¯a. We will use some of this gauge freedom to fix the bulk coordinates such that the brane sits straight at z=0 (in other words, the “brane bending mode” is gauged away). In these coordinates, the bulk metric is related to its boundary counterpart (the induced metric on the brane) as γμν(x)=δμaδνbgab(x,0)=gμν|.(49)As briefly remarked in the previous section, even with the brane frozen at z=0, one can further fix the gauge so that the following conditions hold h¯μz=h¯zz=0.(50)This still leaves some residual gauge freedom: namely, consistently with all previous gauge choices, one can choose a nontrivial parameter ξ¯a=L2(z+L)-2δaμωμ(x), which generates the following transformation of the 5D fields δh¯μν=L2(z+L)2(∂μων+∂νωμ),δVμ=L2(z+L)2ωμ.(51)At the location of the brane z=0, this induces the 4-dimensional gauge transformation of the brane metric δhμν=∂μων+∂νωμ. In addition to hμν, the covariant brane theory (23) contains four Stückelberg scalars ϕμ=xμ+ημνvν, with vμ shifting under the 4D brane diffeomorphisms as δvμ=ωμ(x). At the given (linear) order in fields and gauge paramaters, this shift matches with the transformation (51) of the boundary “image” of the bulk Stückelberg field V¯μ(x,0)=Vμ|. At this order, therefore, we will identify vμ=Vμ|,(52)which should be understood as part of the definition of our theory. This linearized relation will be sufficient for our purposes of showing how a quadratic kinetic term for the longitudinal vμ (that is, the helicity-0 polarization of the 4D graviton) arises from the bulk dynamics.

Before we proceed, it is instructive to recall how counting of d.o.f. works from the point of view of a 4D observer. A massive 5D graviton propagates 9 d.o.f. At energies well above m¯, and in the gauge we are working with, these organize into the 5 d.o.f. of a general-relativistic (helicity-2) graviton, described by h¯μν, and four extra (helicity-1 and -0) d.o.f. that live in the 5D Stückelberg field Va. In the limit m¯→0, the effective 4-dimensional spectrum of the helicity-2 graviton h¯μν(z,x) is well known from previous work on the RS II model [19]. It consists of a gapless continuum of spin-2 KK modes, plus a special, localized zero mode hμν(x). In the theory under consideration, a short-distance 4D observer would additionally see the spectrum of states stemming from the higher-dimensional field Va, which we will study in a little more detail in what follows.

Now, even with a nonzero m¯, the four-dimensional KK spectrum of the higher-dimensional field h¯μν forms a gapless continuum [34]: h¯μν(x,z)=∫0∞dmh¯μνm(x)fm(z),(53)in full analogy with the RS II case (although the KK wave functions fm(z) will be slightly distorted near the origin, compared to the m¯=0 case). On the other hand, the infrared dynamics of the would-be zero mode, hμν=∫0∞dmh¯μνm(x)fm(0), deviates qualitatively from its RS II counterpart. In particular, as is evident from Eq. (51), at energies of order m¯ and lower, hμν(x)=h¯μν| acquires mixing with the Stückelberg vector vμ(x)=Vμ|, eating it up and turning into a long-lived resonance [16,34]. The mass scale of the resonance is set by m¯, and its nonzero, small width (suppressed by extra powers of m¯L compared to its mass [34]) is due to the possibility of decaying into the continuum of KK modes. Taking m¯ of order of the current Hubble rate (and recalling that m¯≪L-1) makes the massive resonance completely stable for all practical purposes.

Having understood the nature of Vμ| as the Stückelberg field enforcing gauge invariance of the 4D effective theory, we proceed to study the bulk dynamics of this field. As discussed around Eq. (13), in the limit m¯→0, Va is a massive AdS vector: SV=M53m¯2∫d5xg¯(-14FabFab-4L2VaVa).(54)We will be particularly interested in the longitudinal part of Va, defined as Va=VaT-∂aΠ (∇aVaT=0). This is because the boundary image π=Π| of this field—the helicity-0 component of the 4D graviton—is responsible for potential strong coupling of the 4D brane theory [16]. On the other hand, the 4D graviton’s helicity-1 component, defined in (27), is related to its bulk counterpart as aμ=VμT|.⁸

We note that the procedure of splitting the bulk field Va into the covariantly transverse and longitudinal components is not unique [16]. Namely, there is a (gauge) reduncancy under VaT→VaT+∇aS, Π→Π+S, where S satisfies □S=0 and is therefore solved, with the appropriate boundary conditions, by S(z,x)=(z+L)2L2K2((z+L)-□4)K2(L-□4)s(x),(55)where by K1,2 we denote MacDonald polynomial. The brane Stückelberg fields of Eq. (27) are given by aμ=VμT| and π=Π| (notice that while VaT is constrained to be 5D transverse, aμ does not have to satisfy any 4D constraint). The above redundancy is then realized on these fields as gauge symmetry under aμ→aμ+∂μs and π→π+s.

Using Eq. (54), one finds for the Π action: SΠ=-4M53m¯2L2∫d5xg¯(∇aΠ∇aΠ).(56)Varying this action yields the dynamical equation □Π=0, which is solved, with the decaying boundary conditions at z→∞, by the following function Π(z,x)=(z+L)2L2K2((z+L)-□4)K2(L-□4)π(x),(57)where □4≡ημν∂μ∂ν and K1,2 are Macdonald functions. Plugging the above solution back into (56) then yields the boundary effective action for the 4D field π(x): Sπ=-4M53m¯2L2∫d4xπ(x)-□4K1(L-□4)K2(L-□4)π(x).(58)Nonlocality of this action is a result of “integrating out” a gapless continuum of KK modes. This can be deduced, for example, by examining the pole structure of the two-point function of π. Nevertheless, at sufficiently low energies corresponding to L-□4≪1, the action (58) can be approximated by a standard, local kinetic term: Sπbdy=4M53m¯2L∫d4xπ□4π.(59)That this approximation is possible is simply an expression of the well-known fact that a massless AdS scalar localizes on a RS II brane, acquiring effectively four-dimensional dynamics at distances greater than L [19]. At the same time, the nonlocality of the action (58) is crucial in that it allows to evade the no-go result, forbidding an independent kinetic term for π in a local and unitary theory of massive gravity [16].

The remainder of this section closely follows Sec. II, with the only difference that here we keep some of the formulas more general and slightly expand on the schematic discussion of that section.

As outlined around Eq. (18), in addition to the π kinetic term (59), the brane-induced action also contains a kinetic term for the helicity-2 field hμν (we stress again that there is also a kinetic mixing between hμν and π, induced from the bulk [16], but we omit this term here as it is unimportant for the interesting range of the parameters of the theory). Adding the brane-induced terms to the original 4D action (29) yields Lbrane+bulk=-MPl24hμν(Eh)μν+4M53m¯2Lπ□4π-M42m24εεh∂2π-M42m24[β2εεh(∂2π)2+β3εεh(∂2π)3]+hμνTμν,(60)where we have defined the effective 4D Planck mass as MPl2=M42+12M53L.(61)The canonically normalized fields are related to those appearing in the action (60) as h^μν=MPlhμν,π^=(M53L)1/2m¯Lπ.(62)With this normalization of the fields, the dimensionless parameter α [defined in Eq. (20)], quantifying the strength of the fifth force mediated by the helicity-0 graviton is α∼M42m2(M42+M53L/2)1/2(M53L)1/2Lm¯.(63)As discussed in Sec. II, for any relevant choice of the parameters, α is an extremely small number (typically of order mL≪1) that tends to zero as the graviton mass m is sent to zero. The linearized theory therefore avoids the vDVZ discontinuity.