PHYSICAL REVIEW D, VOLUME 70, 084028 Massive gravity on a brane
Z. Chacko,1,* M. L. Graesser,2,† C. Grojean,3,4,‡ and L. Pilo5,x 1Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA and Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 2California Institute of Technology, 452-48, Pasadena, California 91125, USA 3Service de Physique The´orique, CEA Saclay, F91191 Gif-sur-Yvette, France 4Michigan Center for Theoretical Physics, Ann Arbor, Michigan 48109, USA 5Dipartimento di Fisica ‘G. Galilei,’ Universita` di Padova, INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy (Received 17 February 2004; revised manuscript received 6 July 2004; published 15 October 2004) At present no theory of a massive graviton is known that is consistent with experiments at both long and short distances. The problem is that consistency with long distance experiments requires the graviton mass to be very small. Such a small graviton mass however implies an ultraviolet cutoff for the theory at length scales far larger than the millimeter scale at which gravity has already been measured. In this paper we attempt to construct a model which avoids this problem. We consider a brane world setup in warped anti- de Sitter spacetime and we investigate the consequences of writing a mass term for the graviton on an infrared brane where the local cutoff is of order a large (galactic) distance scale. The advantage of this setup is that the low cutoff for physics on the infrared brane does not significantly affect the predictivity of the theory for observers localized on the ultraviolet brane. For such observers the predictions of this theory agree with general relativity at distances smaller than the infrared scale but go over to those of a theory of massive gravity at longer distances. A careful analysis of the graviton two-point function, however, reveals the presence of a ghost in the low energy spectrum. A mode decomposition of the higher dimensional theory reveals that the ghost corresponds to the radion field. We also investigate the theory with a brane-localized mass for the graviton on the ultraviolet brane, and show that the physics of this case is similar to that of a conventional four dimensional theory with a massive graviton, but with one important difference: when the infrared brane decouples and the would- be massive graviton gets heavier than the regular Kaluza-Klein modes, it becomes unstable and it has a finite width to decay off the brane into the continuum of Kaluza-Klein states.
DOI: 10.1103/PhysRevD.70.084028 PACS numbers: 04.50.+h from that of Einstein gravity.While there are indications I. INTRODUCTION that suitable ultraviolet completions may be free of the Recently there has been considerable interest in theo- latter problem [14–17] to date no completely satisfactory ries of gravitation which deviate from Einstein’s gravity candidate theories are known [8–11]. at very long distances (for example, [1–5]). However In the absence of a known Higgs mechanism for grav- there has been no consistent theory yet proposed which ity it might seem that these problems pose an insurmount- is consistent with all observations at both macroscopic able obstacle in constructing any experimentally viable and microscopic length scales [6–11]. Conceptually per- theory of a massive graviton. However a closer examina- haps the simplest modification is a mass term for the tion suggests that this need not be the case. Consider the graviton [12]. However this theory suffers from three five dimensional brane model of Randall and Sundrum difficulties which are typical of theories of massive grav- (RS) [18]. This is a simple example of a theory where the ity as a whole. First, unless the mass term has the Fierz- local cutoff varies from point to point in the higher Pauli (FP) form the theory has a ghost [12,13]. Second, dimensional space. In particular in the far infrared the 4 1=5 for a graviton mass mg this theory has a cutoff mgM4 cutoff of the theory is below the millimeter scale, where [7], where M4 is the Planck scale. This cutoff is much too gravity has been measured in the laboratory. This low low for the theory to be simultaneously consistent with cutoff is completely consistent with these experiments experiments at microscopic and macroscopic scales. because physics measurements on the brane at any four Third, even for arbitrarily small graviton mass the lon- momentum scale p are exponentially insensitive to points gitudinal component of the massive graviton does not in the bulk where the local cutoff is lower than the scale decouple from sources. This fact, which was first observed p. by van Dam,Veltman and Zakharov [13], implies that the The success of this theory suggests a means whereby tensor structure of the gravitational interaction deviates the problems normally associated with theories of mas- sive gravity can be avoided. To the single brane model of Randall and Sundrum we add a second brane deep in the *Electronic address: [email protected] †Electronic address: [email protected] infrared such that the compactification radius, which is ‡Electronic address: [email protected] the inverse mass of the lightest Kaluza-Klein (KK) state, xElectronic address: [email protected] is of order galactic size. On this second brane we add a
1550-7998=2004=70(8)=084028(15)$22.50 70 084028-1 2004 The American Physical Society Z. CHACKO, M. GRAESSER, C. GROJEAN, AND L. PILO PHYSICAL REVIEW D 70 084028 brane-localized mass term for the graviton. The extra Kaluza-Klein states, then it becomes unstable and it has a dimension just corresponds to an interval with two finite though small width to decay into the regular boundaries (see Ref. [19] for similar constructions for Kaluza-Klein states. gauge theories and their application for the problem of electroweak symmetry breaking). Since two-point corre- II. BRANE-LOCALIZED FIERZ-PAULI lators with external legs on the ultraviolet brane are ex- MASS TERM ponentially insensitive to physics on the infrared brane for four momenta above the compactification scale we A. Bulk equations of motion expect that conventional Einstein gravity will be repro- We consider a brane world model whose dynamics is duced on the ultraviolet brane at distances shorter than governed by the following action:2 the compactification scale. However, at distances longer Z q R than the compactification scale, the theory is sensitive to S d5x jgj ÿ ÿ L 22 i i infrared physics, with the consequence that below this 5 scale the theory with a Fierz-Pauli mass term is expected p g z ÿ z ; (1) to resemble the four dimensional Fierz-Pauli theory of a 55 i massive graviton. This then would be a concrete realiza- z are the locations of the two branes, is the bulk tion of a an experimentally viable theory of massive i1;2 cosmological constant, is related to the 5D Planck gravity. In this paper we investigate this proposal in de- 5 (fundamental) scale M by M3 1=22, are the brane tail. This picture is correct, at the price of a serious 5 5 5 i cosmological constants (tensions), and are Lagrangian drawback though: while the theory does indeed reproduce Li densities describing some boundary localized matter Einstein gravity at subgalactic length scales the low fields. We will fine-tune the bulk and brane cosmological energy spectrum in the four dimensional effective theory constants such that the background geometry corresponds contains, in addition to a massive graviton, a ghost state. to the well-known Randall-Sundrum solution: A mode decomposition of the higher dimensional theory 1 R 2 reveals that it is the radion field which is a ghost. We also ds2 dxdx dz2 (2) find that this conclusion is rather general: even allowing z for a non–Fierz-Pauli mass term on the IR brane the q ÿ1 2 2 radion field is always a ghost. The analysis of the more with R ÿ5=6, UV 6= 5R and IR general case is provided in Appendix B. ÿUV. The location of the branes are such that the In the following sections we explore in detail the model warp factor R=z is set to 1 on the ultraviolet (UV) brane we are investigating. We compute the graviton two-point (z1 R), and it is exponentially smaller on the IR brane 0 function with external legs on the ultraviolet brane and (z2 R R). show that while the predictions of the theory agree with The aim of this paper is to study the spectrum of the those of general relativity for observers on the ultraviolet physical excitations when nontrivial gravitational inter- brane probing distance scales shorter than the compacti- actions are introduced on the branes. We thus need to fication radius, the light states consist of a massive gravi- consider gravitational fluctuations around the RS back- ton and a ghost. We then perform a mode decomposition ground solution: of the linearized theory for both the transverse traceless ds2 e2A h dxMdxN; with A ÿln z=R: modes and the radion. This reveals that the transverse MN MN traceless modes of the theory without a mass term (3) smoothly go over to the transverse traceless modes of In the bulk, the Einstein’s equations are of course the theory with a mass term as the mass term is turned on. independent of brane interaction terms. At the linear level However the same is not true of the radion. Instead, the and in absence of any matter beside the bulk cosmological radion changes discontinuously into a ghost as soon as the constant, these equations read mass term is turned on. We also study the theory with a 1 1 2 2A mass term on the ultraviolet brane and show that the EMN GMN 5e hMN 0; (4) predictions of this theory for observers on the ultraviolet 1 where GMN is the linear piece of the Einstein tensor. brane agree with those of a four dimensional theory with Using the Einstein equations of the background solution, a massive graviton. However there is one important dif- we finally arrive at ference: if the extra dimension is sufficiently large that the would-be graviton is heavier than the lightest regular 2Our conventions correspond to a mostly plus signature ÿ ... and the definition of the curvature is such that a 1Under certain circumstances theories with ghosts may in Euclidean sphere has a positive curvature. Bulk coordinates fact be viable [20,21]. However we do not pursue this possi- will be denoted by capital Latin indices and brane coordinates bility here. by Greek indices.
084028-2 MASSIVE GRAVITY ON A BRANE PHYSICAL REVIEW D 70 084028 1 1 1 3 E 1 @ @h @ @h ÿ ᮀh ÿ @ @ hÿ @ @ h ÿ ᮀh ÿ h00 ÿ h00 ÿ h0 ÿ h0 A0 2 2 2 2 1 1 1 3 ÿ @ @ h ᮀh ÿ 3A00h ÿ 3A0h0 ÿ 9A02h @ h0 @ h0 ÿ@h0 A0 @ h 2 55 2 55 55 55 55 2 5 5 5 2 5 0 @h5ÿ3A @ h5; (5)
1 1 0 3 0 1 for which the 4D fluctuations are TT, i.e., traceless, h 0, E5 @ h ÿ @h A @h55 @@ h5 3 2 2 2 and transverse, @ h 0. This gauge will be denoted 1 GNTT. Each gauge has its own advantage: in the GNTT ÿ ᮀh 3A00h ÿ 3A02h ; (6) 2 5 5 5 gauge it will be easy to solve the bulk equations of motion while in the GNUV and GNIR gauges it will be easy to 1 3 E 1 ÿ @@h ÿ ᮀh A0h0 ÿ 6A02h solve the boundary conditions. In the following sections 55 2 2 55 we will explain how these different gauges are related to 0 ÿ 3A @ h5; (7) each other depending on the interactions and the matter localized on the branes. Finally note that the GNUV and with the following conventions: 4D indices are raised and GNIR gauges still possess usual 4D reparametrization lowered using the flat Minkowski metric, h is the 4D trace invariance associated to the x. h, ᮀ @ @ and a prime denotes a derivative with respect to the z coordinate. The bulk equations are obviously covariant under an B. Boundary conditions in presence infinitesimal general coordinate transformation that reads of brane mass term at the linear order: We now want to add some brane-localized interactions for the gravitational degrees of freedom. More precisely, xM M; (8) we are interested in a localized mass term.Working in the generalized Gaussian system of coordinates in which the h ÿ@ ÿ @ ÿ A05 ; 2 (9) brane we want to add the mass term on is straight, the 0 5 5 A 0 ÿA h5 ÿ ÿ @ ;h55 ÿ2 e e : mass term is for simplicity chosen to be of the Fierz-Pauli form4: Clearly, this reparametrization invariance allows to re- Z strict ourselves to generalized Gaussian normal (GGN) 1 1 R4 L ÿ f4 d4x h h ÿ h2 ÿ f4 systems of coordinates: 8 UV jzR 8 IR R04 Z h55 h5 0; brane embeddings : z fi x; 4 2 d x hh ÿ h jzR0 : (14) i 1; 2: (10) Of course, since the mass terms (14) explicitly break Within these generalized Gaussian normal gauges, there general coordinate invariance, their forms will not be is still a residual reparametrization invariance involving the same in a different system of coordinates and it will arbitrary functions, and , of the 4D coordinates: have to be determined by coordinate transformation from z 5 5 x; z x; (11) the appropriate GGN gauge. Note, in particular, that the R two mass terms are not written in the same coordinate systems: the UV mass term is written in the GNUVgauge 1 z2 x; z xÿ @ x; (12) while the IR mass term is written in the GNIR gauge. 2 R and the transformation of the metric fluctuations is 3To see this, first note that the residual gauge invariance can be used to set h 0 and @ h 0 at a point z z0. Then the z2 2 h @ @ ÿ @ ÿ @ : (13) 5 equation implies that @h ÿ @ h 0 everywhere in the R R bulk. Using this result the (55) equation then implies h0 0 everywhere. But since h 0 at a point, it is zero everywhere. Clearly, with an appropriate choice of we can maintain Then @ h 0 in the bulk. the GGN gauge fixing conditions (10) and straighten one 4The powers of the warp factor are determined by the of the branes which will now be located at a constant z. requirement that in the coordinate system where eA 1 at This gauge choice for which the UV(IR) brane is straight the IR brane say, the boundary condition on that brane is independent of the warp factor on the UV brane. will be called (GNIR)GNUV, generalized Gaussian nor- 5In an abuse of language we will still refer to generalized mal gauge with respect to the UV(IR) brane. There is coordinate transformations as ‘‘gauge transformations,’’ even finally a third special generalized Gaussian normal gauge though this symmetry is explicitly broken.
084028-3 Z. CHACKO, M. GRAESSER, C. GROJEAN, AND L. PILO PHYSICAL REVIEW D 70 084028 More general, non–Fierz-Pauli mass terms may also be C. State counting considered. For brevity, the analysis of these more general In this section we count the number of degrees of mass terms is provided in Appendix B. The conclusions freedom in the gravity theory with a brane-localized of this and subsequent sections is unchanged in the more Fierz-Pauli mass term. In the theory without such a general case: an additional state is present, but it decou- mass term the spectrum consists of a massless spin-2 ples and the radion is always a ghost. field with two polarizations, a radion and a tower of The effect of the mass term and additional matter massive spin-2 Kaluza-Klein resonances with five polar- localized on the brane is to modify the usual Neumann izations each. Here we show that once the mass term is boundary condition for the metric fluctuations. The new introduced the spectrum changes only in that the lightest boundary condition gets simplified on the GGN gauge spin-2 field is now massive and therefore has the five where the brane is straight. In the GNUV gauge, the polarizations associated with a massive spin-2 particle. boundary condition at the UV brane is In doing so we will make use of the important result 1 that in the presence of the brane mass term and in the h0 ÿ2 SUV ÿ SUV ÿ f4 h ; (15) absence of any additional matter on the massive brane, the 5 3 UV gauge GNIR(GNUV) is equivalent to the GNTT gauge. In UV subsequent sections we will also make use of this result. where S is the stress-energy tensor for the matter localized on the UV brane, SUV is its trace, SUV. Before demonstrating this, we note that this is similar to the situation with a massive U 1 vector boson A . Similarly the boundary condition at the IR brane is There one has no gauge invariance, but a priori 4 degrees of freedom. However, the equations of motion for the 0 2 IR 1 IR 2 4 R h S ÿ S ÿ f h (16) 5 3 5 IR R0 vector boson imply that @A 0, eliminating one of the unphysical perturbations. Thus the theory describes where warp factors have been absorbed into our definition three fluctuating degrees of freedom, the correct number IR of S. These boundary conditions are obtained by vary- for a massive spin-1 particle. ing the brane-localized mass term action given above, Similarly, a massive graviton in four dimensions a and then adding that to the left-hand side of the linear- priori describes 10 degrees of freedom, but has no gauge ized Einstein equations. The boundary conditions are invariance. As with the massive vector boson, one finds then obtained by requiring the cancellation of the bound- that for the Fierz-Pauli mass term, the equations of mo- ary terms in the variation of the action (see for instance tion imply that the metric is transverse and traceless. This [19] for an analogous computation in gauge theories). For eliminates five perturbations leaving five, which is the later convenience it is useful to introduce the following correct number for a massive spin-2 particle [12,22]. parameters that have the dimension of a mass The situation with the brane-localized mass term is similar, but naively worse. This is because we are describ- 2 4 IR UV 5fIR UV: (17) ing a five dimensional gravitational theory, which a priori has 15 degrees of freedom. Since the brane mass term We can find the graviton Kaluza-Klein spectrum by explicitly breaks general coordinate invariance, there is a solving the bulk equations for transverse and traceless concern that additional states which were previously excitations, supplemented by the above boundary condi- eliminated by the gauge invariance are now reintroduced. tions in the absence of matter on the branes. The presence This would be a disaster for the model, just as for a of the brane mass terms lifts the zero mode from the massive graviton theory with a mass term in a spectrum. For instance, in the presence of a mass term non–Fierz-Pauli combination. ÿ1 on the IR brane only and in the limit IR R ,the However, just as in the massive vector and graviton lightest spin-2 state has a mass given by (see Sec. IVA for examples given above, we shall see that the equations of details) motion imply that for a Fierz-Pauli mass term defined in GNIR(GNUV) gauge, the metric is additionally TT in 2 4 4 4 2 5fIR R 2 R absence of any additional matter on the brane. Thus the m0 2 0 2 0 fIR (18) R R MPl R only degrees of freedom are massive gravitons which involve only five physical polarizations and a massless 2 2 where MPl R=5 is approximately the four dimensional radion associated with the movement of the brane. In Planck mass. addition, since this result will follow from the properties The spectrum of light states with mass below the of the bulk equations of motion and the IR(UV) boundary effective compactification scale 1=R0 is seen to also con- condition, these conclusions are unchanged if a source is tain a massless scalar, the radion. Part of this paper is placed on the opposite UV(IR) brane. devoted to the identification of the properties of this To see this, let us consider the case of a mass term perturbation. added on the IR brane and let us work in the GNIR gauge
084028-4 MASSIVE GRAVITY ON A BRANE PHYSICAL REVIEW D 70 084028 Z q where the IR brane is straight. The IR boundary condition 4 0 0 PQ 0 0 0 hMN x; z d x dz jgjMN X; X SPQ x ;z; (20) for the combination H @ h ÿ @h is
R where is the Green’s function. We are mostly interested @zHjzR0 ÿIR 0 HjzR0 : (19) in physics for an observer on the Planck brane and so we R want to compute the two-point correlator with both ex- ternal legs on the UV brane. Finding this correlator is Now the 5 equation implies that @ H 0, so the left- z equivalent to computing, in the GNUV coordinates, the hand side of the equation above vanishes identically. Since metric perturbation on the UV brane as a response to a IR 0, we learn that H vanishes at the location of the source localized on the UV brane too. IR brane. But since by the 5 equation H is constant in To obtain the two-point function we closely follow the the bulk, we find that it actually vanishes identically. work of Garriga and Tanaka [24]. The approach is to Using this result the (55) component of the bulk determine, at a point on the UV brane, the linearized Einstein equations implies that @zh 0 identically. gravitational field created by a source on the same UV Next consider the IR boundary condition again, but writ- brane. This can be related in a simple way to the graviton 0 ten as @zhjzR0 ÿIRR=R hjzR0 . The trace of this two-point correlator with external legs on the UV brane. 0 implies @zhjzR0 ÿIRR=R hjzR0 , which when com- The key observation is that it is convenient to first work in bined with the previous result implies that hjzR0 0. the GNTT gauge where the bulk equations are very sim- But @zh 0,soh vanishes in the bulk. From this result ple. In this gauge the equations in the bulk reduce to and H 0, it follows that @ h 0 identically. 2 3 The introduction of a brane-localized Fierz-Pauli mass ᮀ @z ÿ @z h 0: (21) term (14) therefore implies that the metric is transverse z and traceless in the GGN gauge where the brane is straight However, in this gauge both branes will in general not be (the GNIR gauge is also GNTT). Thus at the massive level straight and the bending of each brane provides an addi- there are only 5 degrees of freedom, corresponding to the tional contribution to the stress tensors on the two helicity states of a massive graviton. The brane-localized boundaries, modifying the boundary conditions. The Fierz-Pauli mass term does not introduce any additional main effort of these sections is to determine this modi- massive degrees of freedom that were not already present fication. With that information and the solution to the in the RS model. propagator in the GNTT gauge, we can readily evaluate These results may be understood by noting that the the perturbation in the GNIR(GNUV) gauge . model still has a large residual general invariance gen- We consider below two cases. In the first example, both erated by coordinate transformations that vanish at the the Fierz-Pauli mass term and the source are located on location of the brane. Referring to (9), this requirement the UV brane, and the perturbation on the UV brane is 5 implies jzR0 0 and jzR0 0. Thus the only gauge determined. In the second, the Fierz-Pauli mass term is transformations explicitly broken by the brane mass term placed on the IR brane, with the source still placed on the are those associated with the would-be zero mode gravi- UV brane. ton and the bending of the brane. A. Fierz-Pauli mass term on the Planck brane III. TWO-POINT FUNCTION ANALYSIS In GN coordinates around the Planck brane (GNUV), the UV boundary condition is In this section we obtain expressions for the graviton two-point correlator with external legs on the UV brane.6 1 @ hGNUV hGNUV ÿ 2 SUV ÿ SUV : (22) We consider first the case with a mass term on the UV z UV 5 3 brane and then the case with a mass term on the IR brane. This calculation serves two purposes. We will be able to with determine the extent to which observers on the UV brane 2 4 UV f : (23) find the theory to deviate from Einstein’s gravity at any 5 UV particular length scales.We will also be able to determine In order to be able to solve the Einstein’s equations in the the masses of the light modes in the four dimensional bulk, it is useful to perform a coordinate transformation effective theory. This is precisely the physics we are most in order to obtain a graviton perturbation that is trans- interested in determining. verse and traceless. The transformations (11) and (12) to The metric perturbation in the linear approximation TT coordinates yields a new boundary condition: created by a source S will be given by GNTT GNTT 2 @zh UVh ÿ 5 (24)
6The analogous calculation for the case of a gauge field with where the source term now includes a brane-bending a brane-localized mass term may be found in [23]. contribution:
084028-5 Z. CHACKO, M. GRAESSER, C. GROJEAN, AND L. PILO PHYSICAL REVIEW D 70 084028 1 2 2 GNUV 25 1 1 1 S ÿ S ÿ @@ h ÿ S ÿ S 2 0 2 ᮀ 3 5 R 1 ÿ R=R 2 UV 2 2 R@ @ ÿ @ ÿ @ : R 2 1 2 R ÿ 5 S: (32) 5 3R R0 1 ÿ R=R02 ᮀ (25) The gauge parameters and are chosen so that in the The crucial factor of 1=3 from the gauge transformation has been combined with the 1=3 factor appearing in the new frame the metric is TT. Since is the source for the metric perturbation in the GNTT gauge, it must be trace part of the Green’s function to obtain the correct transverse and traceless too. This leads to the two con- factor of 1=2 for a massless graviton [24]. The part that is ditions below on the gauge parameters: left over is interpreted as due to the exchange of the radion, and appears here with the correct sign to describe 6 a physical propagating particle. i @ ÿ @ @ ᮀ 0; (26) UV R The important point in this review of the results of Garriga and Tanaka is to draw attention to the technical reason for recovering the correct tensor structure of the 2 8 5 ᮀ ᮀ iiÿ S ÿ 2 UV R ÿ 2@ 0: massless graviton: the transformation between the TTand 3 R GN coordinate system involved a nonvanishing bending (27) of the brane ~5 . By contrast, a transformation involving can only modify the part of the graviton propagator involving derivatives of the source, leaving the part in- 1. Massless case: 0 UV volving the trace untouched. This is the situation encoun- To begin though, first suppose that no mass term is tered when, on the brane, a graviton mass term is turned present. Then we should recover the results of Garriga and on. Tanaka [24]. In this case the TT conditions simplify and ᮀ ÿ 2 reduce to the single requirement that 5S=6.In 2. Massive case: 0 the GNTT coordinates, the source is thus related to the UV brane stress-energy tensor by: For a nonzero mass term on the brane the first require- ment (26) becomes nontrivial and implies that 1 @ @ S ÿ ÿ S (28) 3 ᮀ ᮀ 0: (33) which is manifestly transverse and traceless. The solution for the metric fluctuations in the bulk is: Decomposing the vector into a scalar and a transverse T T Z part, ÿ @ with @ 0, the second condi- GNTT 2 4 0 0 0 tion (27) relates the scalar part to the brane stress-energy h x; zÿ d x x; x ; z; R x (29) 5 tensor: where is the Green’s function for a scalar field in the 2 4 Randall-Sundrum background. It satisfies the boundary ᮀ 5 S ÿ : (34) conditions @ @ 0 and its solution may be 6 R z jzR z jzR0 UV found in the appendices. Back in the GN system the metric perturbation on the brane is A consistent solution to the TT conditions is to set T 2 0. This leads to the same expression (28) for the 5 1 GNUV GNTT source in terms of the boundary stress-tensor S. h x; R h x; R ᮀ S: (30) 3R Crucially though, the coordinate transformation needed where we have substituted for and dropped terms in- to reach the GNTT frame now involves rather than . volving longitudinal four dimensional derivatives. At Thus there is no brane bending to compensate the 5D long distances [see AppendixA, Eq. (A10)], q Rÿ1, structure of the brane propagator and we expect that the propagator becomes gravity is never Einsteinian on the brane. Going back to the GNUV frame using 0 2 1 1 4 0 x; R; x ;R! 0 2 x ÿ x (31) R 1 ÿ R=R ᮀ GNUV GNTT h h ÿ 2@@ ; (35) and the metric perturbation in the GNUV coordinate is then [24] (again dropping terms involving longitudinal we get a different expression for the metric fluctuation four dimensional derivatives) compared to when no brane mass term is present:
084028-6 MASSIVE GRAVITY ON A BRANE PHYSICAL REVIEW D 70 084028 Z m & H , the lifetime is much longer than the age of the hGNUV x; zÿ2 d4x0 x; x0; z; R S 0 5 universe. Finally, one may be puzzled by the absence of any term 1 @ @ 2 @ @ 5 in (36) that could possibly be interpreted as due to the ÿ ÿ ᮀ S ÿ ᮀ S: 3 3UV exchange of a radion with nonderivative couplings. As (36) shown at the end of Sec. IV, the radion is normalizable and physical (not a ghost), and has a wave function that is Here is the Green’s function for a scalar field in the localized about the IR brane. However, in contrast to RS, Randall-Sundrum background with a mass term on the here the radion wave function vanishes at the UV brane UV brane. It satisfies the boundary conditions @ z jzR and has only derivative couplings to sources located there. and @ 0. Its expression is given in the UV jzR z jzR0 Therefore it does not contribute to the two-point correla- appendices. tion function of two conserved sources located at the UV As already mentioned, unlike the case with no brane brane. mass term, here there is no brane bending. Thus the trace part of the propagator is the same in the GNTT and B. Fierz-Pauli mass term on the infrared brane GNUV coordinate systems; in particular, the factor of 1=3 does not change and Einstein gravity is not recovered. We consider the case where the Fierz-Pauli mass term In order to decouple the IR brane (R=R0 ! 0)while is on the IR brane. The source remains on the UV brane. The first observation is that in the GGN coordinate keeping fUV held fixed, we consider the long distance limit qR 1 but keeping qR0 1 in order to probe the system with respect to the IR brane (GNIR), the metric fifth dimension. Using the approximate expression (A12) is additionally TT, since there is no source on the IR brane of the propagator found in AppendixA for this limit, we (see Sec. II). Thus the metric satisfies (21) in the bulk, arrive at with the IR boundary condition R 2 GNTT GNTT 2 1 @zh ÿIR h ; (38) hGNUV x; Rÿ 5 R0 R ᮀ ÿ m2 with 1 @@ S ÿ ÿ 2 S ; (37) 2 4 3 m IR 5fIR: (39)
2 In this gauge however the UV brane is bent, due to the where m 2UV=R. This is the correct propagator for a massive spin-2 particle [22], up to and including the source located there. To determine the UV boundary derivative terms that scale as mÿ2. At distances R r condition in the GNTT gauge, we first consider the GN R0, it is not surprising then to find that the perturbation is coordinates with respect to the UV brane (GNUV) and dominated by the exchange of a single massive spin-2, then perform a coordinate transformation. In the GNUV gauge, the UV boundary condition is with the exchange of the KK tower suppressed as in the Randall-Sundrum model. 1 @ hGNUV ÿ2 S ÿ S : (40) At energy scales much below the compactification z 5 3 scale, r R0, the theory is four dimensional and the only light states are the radion and a massive graviton, Inserting the transformations (11) and (12) relating the for which there will be a van Dam-Veltman-Zakharov GNUV gauge and GNTT gauge metric into the above discontinuity. But in the limit just considered, where the boundary condition gives the desired UV boundary con- IR brane is decoupled first, we cannot appeal to these dition in the GNTT gauge: arguments, as the theory is never four dimensional. GNTT 2 @zh ÿ ; (41) Nevertheless, the result above, (37), demonstrates that 5 even in this intrinsically five dimensional limit there is where the source term now includes a brane-bending still a discontinuity. contribution: Since in this limit there is a mass gap between the 1 would-be zero mode graviton and the continuum of bulk S ÿ S ÿ 2ÿ2@ @ : (42) 3 5 gravitons that goes down to zero, following the reasoning of [25] one may suspect that the graviton studied here is As before, requiring that this source is transverse and ᮀ 2 unstable. In the appendices the two-point function is traceless fixes ÿ5S=6. In GNTT gauge then, the evaluated for complex, timelike q2. There we find that UV source is the light graviton studied above does have a complex 1 @@ pole, with a lifetime ÿ given by ÿ=m mR2.Thislife- S ÿ ÿ S: (43) 3 ᮀ time is parametrically identical to the scalar example studied in [25]. Since here though Rÿ1 * 10ÿ3 eV and The solution for the metric perturbation in the bulk is
084028-7 Z. CHACKO, M. GRAESSER, C. GROJEAN, AND L. PILO PHYSICAL REVIEW D 70 084028 Z Finally, we can see that for UV brane observers hGNTT x; zÿ2 d4x0 x; x0; z; R x0; (44) 5 Einstein gravity is recovered at distances shorter than the compactification scale. Consider qR0 1 but qR where satisfies the Green’s function equation 1. In this limit all dependence on the IR brane disappears. 3 z3 Using results (A11) obtained in AppendixA, the pertur- ᮀ @2 ÿ @ x; z; x0;z0 4 x ÿ x0 z ÿ z0; z z z R3 bation on the UV brane is indeed found to be (still dropping terms involving longitudinal 4D derivatives) (45) 2 GNUV 25 1 1 with boundary conditions @zjzR 0 and @zjzR0 h x; Rÿ S ÿ S : (50) 0 R ᮀ 2 IR R=R jzR0 0. The Green’s function is given in the appendices, and more details can be found there. Nevertheless when fIR 0, the theory has a ghost which Back in the GNUV coordinate system, the metric per- is responsible for the recovery of 4D gravity on the Planck turbation is brane. We will show in Sec. IV that the ghost mode is the radion. 2 1 hGNUV x; RhGNTT x; R 5 S; (46) 3R ᮀ IV. MODE DECOMPOSITION ANALYSIS where we have substituted for and dropped terms in- volving four dimensional derivatives. Using results (A7)– A. Spin-2 excitations: graviton mass spectrum (A9) obtained in AppendixA, at long distances where we We are first interested in the spectrum and the KK cannot probe the KK excitations, qR0 1,theasymp- decomposition of the spin-2 excitations. In the GNTT totic form of the propagator is gauge, the bulk equations of motion do not couple the different polarizations and thus simply reduce to the N x; R; x0;R! 4 x ÿ x0; (47) scalar equation of the form: ᮀ ÿ m2 3 where and m2 may be found in the appendices. ᮀ 00 ÿ 0 0: (51) N z Focusing on nonderivative terms, in this limit the GNUV metric perturbation reduces to The mode decomposition can be written as X N 1 z 2 GNUV 2 x; z n z n x; (52) h x; Rÿ5 2 S ÿ S ᮀ ÿ m 3 n R 2 5 1 the wave functions n z then satisfy a Bessel equation of S: (48) 3R ᮀ order 2 (mn is the 4D mass of the eigenmode): The last term is due to the gauge transformation between 1 4 00 0 m2 ÿ 0; (53) the GNTT and GNUV coordinates and is independent of n z n n z2 n the IR boundary mass term. In this limit the first two terms describe the exchange whose solutions are of a massive graviton, and the last term describes the m zAnJ2 mnzBnY2 mnz; (54) exchange of a massless scalar (the radion). But the sign of the last term implies that the radion is a ghost. This where the two constants are fixed by the boundary and the conclusion is independent of the size of the IR Fierz- normalization conditions. The boundary conditions for the spin-2 excitations are unaffected by coordinate trans- Pauli mass, fIR. An interesting limit to look at is when 4D momenta can formations of the form (11) and (12) and therefore take the only probe the lightest graviton and not the regular KK same form within the GNTT gauge as in the GNUV and excitations: m q 1=R0, then N ! 2=1 ÿ GNTT gauges: 0 2 R=R =R, and the metric perturbation on the UV brane 0 2 becomes (again dropping terms involving longitudinal n ÿ UV n 0; (55) R jzR 4D derivatives) 2 0 2 R 5 1 1 1 : hGNUV ÿ2 S ÿ S n IR 0 n 0 (56) R 1 ÿ R=R02 ᮀ 2 R R jzR0 2 2 Clearly as soon as a nonvanishing mass term is turned on R 1 ÿ 5 S; (49) at either brane, the would-be massless mode, R2=z2, 3R 1 ÿ R=R02 R02 ᮀ cannot satisfy the boundary conditions: the massless with O IRR corrections not included. This expression mode gets lifted by the brane-localized masses. For the goes smoothly over to the result for Randall-Sundrum. massive modes, the boundary conditions (55) and (56)
084028-8 MASSIVE GRAVITY ON A BRANE PHYSICAL REVIEW D 70 084028 0 2 lead to the quantization equation: when UVR R=R , then the mass of the lightest graviton is approximated by m J m Rÿ J m R n 1 n UV 2 n m Y m Rÿ Y m R 0 2 n 1 n UV 2 n 2 R 2 0 R 0 m0 UVR 02 : (64) mnJ1 mnR IR R0 J2 mnR R R 0 R 0 : (57) m Y m R 0 Y m R n 1 n IR R 2 n The mass of this mode can become larger than the Let us examine the solutions of this quantization equation compactification scale when 1=R0 gets smaller and 0 in the two special cases when a single brane mass term is smaller and UVR held fixed. In the limit 1=R ! 0,it turned on at either the IR or the UV brane. becomes non-normalizable and is no longer in the spec- trum. Instead, a resonance with a finite lifetime is found (see Appendix 3). 1. Mass term on the IR brane UV 0;IR 0 Assuming that m R0 1 and expanding the Bessel n B. The radion as a ghost functions near the origin, we find that the lightest mode has a mass approximately given by 1. The radion wave function 8 R R 2 To provide further evidence that the interpretation of m2 IR : (58) 0 R02 4 R R0 the two-point function obtained previously is indeed cor- IR rect, in this section the radion’s wave function is deter- There is a gap of order R=R0 between this lowest mode mined and its effective action computed. The principal and the regular KK modes that have mass result of this section is a confirmation that when the Fierz- x Pauli mass term is on the IR brane the radion is a ghost. m n (59) n R0 This conclusion is unchanged even if we allow for a non–Fierz-Pauli mass term on the IR brane. The details of where xn n 1=4 are the roots of the J1 Bessel that analysis are provided in Appendix B. function: J1 xn0. In fact, the wave function is rather straightforward to The normalization of the wave function can be found obtain in the GNTT coordinate system. By Lorentz co- analytically using the Wronskian method (see for in- variance the metric describing massless scalar fluctua- stance Ref. [26]). We found tions i must be proportional to @ @ i with ᮀi 0. J m R Inspecting Einstein’s equations in GNTT coordinates the z J m zÿ 1 n Y m z ; n mn N n 2 n 2 n general solution is trivial to obtain. It is : Y1 mnR (60) z4 hGNTT ÿ @ @ f @ @ ; (65) with 2R3 2 2 2 02 1 2 IRR 4IRR mnR where f and are massless scalars. At this point the 2 2 2 0 0 0 2 N n mnR IR RY2 mR mnR Y1 mR boundary conditions are not yet imposed, since the branes 1 are in general bent. For future reference, in this system the ÿ 2 : (61) Y mnR UV and IR branes are located at zUV R ÿ x and 1 0 0 zIR R ÿ R =R 2 x (the normalization is chosen for later convenience). 2. Mass on the UV brane IR 0;UV 0 As a check on this result, the wave function of For a large warp factor, R0=R 1, the quantization Charmousis, Gregory and Rubakov (CGR) [27] for the Eq. (57) can be approximately simplified to radion is now recovered. To do this, transform to the GNUV coordinate system z GNUV z z=R where m Y m Rÿ Y m RJ m R00; (62) n 1 n UV 2 n 1 n the UV brane is straight and located at zUV R.The 0 0 the solutions of which form the regular KK modes again IR brane is located at zIR R ÿ R =R x with 2 ÿ . The new metric is obtained from the roots, xn,oftheJ1 Bessel function: x n z4 z2 2 mn 0 : (63) hGNUV ÿ @ @ f @ @ @ @ : R 2R3 R R On top of this tower, there is another mode that is con- (66) tinuously connected to the massless graviton when UV GNUV goes to zero. For this special mode to be parametrically The UV boundary condition in this system is @zh lighter than the regular KK modes, the mass term added 0 and determines the unknown function to be f on the UV brane must be small enough. More precisely, with still unconstrained. This gives the CGR solution
084028-9 Z. CHACKO, M. GRAESSER, C. GROJEAN, AND L. PILO PHYSICAL REVIEW D 70 084028 z4 z2 2 the IR brane is straight and the radion wave function is hGNUV ÿ @ @ f f @ @ : 2R3 R R given by (68), and perform a final coordinate transforma- tion of the form (67) zF z Actually this is not quite the CGR solution, for here there z rad z ÿ f x (69) is the additional term proportional to . In the RS1 model R with no brane mass term, this term can be gauged away. maintaining h5 0 but not h55 0. F is an arbitrary This is because in the restricted GN coordinate system function with the only restriction that both branes are now with the Planck brane straight and no mass term on any straight, which implies F R00 and F Rÿ1.The brane, there is a residual gauge invariance given by normalization of the radion wave function will be found @ =2 that may be used. to depend only on the values of F at the location of the When the IR brane mass term is present this gauge branes, and not on its particular shape. The metric in this invariance does not exist and cannot be eliminated in final coordinate system is this way. It is seen below that in unitary gauge this mode 2 2z h rad c z@ @ f ÿ F zf ;h rad F0f; is eliminated by the IR boundary condition. In a nonuni- R 55 R tary gauge corresponds to the Goldstone boson asso- (70) ciated to the longitudinal component of the graviton. with Z To determine the radion function in the GNIR coordi- z4 z z0 c zÿ ÿ 2 dz0 F z0: (71) nate system with the IR brane straight, it is useful to recall 2R3 R the following result, derived previously in Sec. II. It is straightforward to verify that for arbitrary F,re- Namely, when the IR brane mass term is present the stricted to the boundary conditions F R00 and metric in these coordinates is in addition TT. But the F Rÿ1, this expression satisfies the equations in the most general solution of Einstein’s equations for a mass- bulk and also both boundary conditions. less scalar in GNTT gauge was already obtained. It is An inspection of this solution indicates a significant z4 hGNIR hGNTT ÿ @ @ f @ @ : (68) difference between the radion here and in the RS1 model. 2R3 Here the brane mass term forces the non-TT part of the (Compared with previous notation here we have defined radion wave function to vanish at the IR brane. That is, 0 slightly differently and pulled out a factor ). This con- here F R 0. This is equivalent to the requirement that tains two massless scalars. One of these is the radion and the metric be traceless in the GNIR coordinate system. the other, as mentioned above, is the Goldstone boson This fact is instrumental in turning the radion into a corresponding to the longitudinal component of the ghost. graviton. In the unitary gauge the IR boundary condition To determine the radion kinetic term we want to inte- GNTT 0 GNTT grate out the short-distance variation of the metric. To do is @zh ÿIR R =Rh . This determines in terms of f, or equivalently, determines after setting this we follow the methodology of [6,28]. For full details, 0. such as carefully adding the Gibbons-Hawking boundary In a nonunitary gauge is no longer zero. But the IR term and seeing that the massive gravitons decouple, or boundary condition is then modified due to an extra term for the more general case of a non–Fierz-Pauli mass term, coming from the Goldstone boson, which may be identi- see Appendix B. Here we quote the main results for the fied with . case of the Fierz-Pauli mass term. Expanding the action In summary, in GNIR gauge (68), with 0,isthe to quadratic order gives radion wave function. Z 3 1 4 R AB Seff ÿ 2 d xdz h EABhCD 45 z 2. The radion kinetic term boundary terms brane mass term: (72) The wave function obtained above is not very useful for determining the effective action, since the UV brane is Next we insert the expression (70) for the radion into the not straight.Wewould like to compute the effective action action, without using its four dimensional equations of in a coordinate system with both branes straight. motion. Since nonderivative terms in the wave function To do this, begin in the GNIR gauge. The radion wave satisfy Einstein’s equation, we are guaranteed that the function is given by (68), and the UV brane is located at integrand is of the form fᮀf. It is then a matter of z R ÿ x. We first need to find the position of the UV collecting terms appearing in the linearized Einstein brane in the GNIR gauge. But this has been determined equations which have four dimensional derivatives. 7 already, since here the UV boundary condition is the Then we find same as in RS1. Hence Eq. (66) and (67) yield f. Next we straighten both branes. To do this, it is easiest 7We have added in the boundary terms. For more details see to start again in the GNIR GNTT coordinates where Appendix B.
084028-10 MASSIVE GRAVITY ON A BRANE PHYSICAL REVIEW D 70 084028 Z Z 1 R0 R 3 see (70)—vanishes on the UV brane since in this model S ÿ d4x dz hABE h eff 2 AB CD F R0. 45 R z Z These conclusions generalize to the case with a 1 R 3 4 0 non–Fierz-Pauli mass term. Here we summarize the re- brane mass term ÿ 2 d x h h 85 z sults of Appendix B. If the mass term is on the IR brane, Z 3 the radion is still a ghost but now there is an additional 0 R0 3 4 R R0 ÿhh jR 2 d x 4 h55hjR state that is decoupled and has a physical kinetic term. 85 z Z Z If the mass is on the UV brane, the radion still has a 3 R0 4 ᮀ 0 physical kinetic term, but now there is an additional state ÿ 2 d xf f dzF 25R R that is a ghost. Both of these results are not surprising Z 3 from the perspective of the AdS/CFT correspondence. 0 4 ᮀ ÿ 2 F R ÿF R d xf f (73) 25R V. CONCLUSIONS Since F R00 and F Rÿ1, the radion kinetic term is We have investigated the physics of brane-localized mass terms for the graviton in warped backgrounds. We Z 3 have performed a linearized analysis of the graviton two- S ÿ d4xfᮀf (74) eff 2 point correlator as well as a mode decomposition of the 25R five dimensional theory. We find that if the mass term is which has the wrong sign. The radion is a ghost! localized on the UV brane, observers on that brane see Repeating this calculation for RS1 provides an inde- physics similar to that of a massive graviton in four pendent check on the overall sign, since here the radion is dimensions. One important distinction, however, is that known to be healthy. In fact, the formula is the same and if the graviton mass is larger than the mass of the lightest the boundary condition in the UV is the same, but the IR Kaluza-Klein modes it can now decay off the brane into boundary condition is different. So FRS1 Rÿ1 and these states. 0 02 2 FRS1 R ÿ R =R , implying that the radion has a A Fierz-Pauli mass term for the graviton on the IR physical kinetic term. brane reproduces Einstein’s gravity for observers local- It is straightforward to repeat this exercise when the ized on the UV brane at length scales shorter than the Fierz-Pauli mass term is on the UV brane. The radion inverse mass of the lightest Kaluza-Klein modes. At wave function in the GNUV coordinates is still given by length scales longer than this the spectrum consists of a (68), but here the integration constant is different in massive graviton and a ghost. It is the radion field which is order to satisfy the UV boundary condition. Just as in the the ghost. previous example, here it is the IR boundary condition For a non–Fierz-Pauli mass term on the IR brane there that determines the position of the IR brane in the GNUV is an additional, physical state in the theory. But the 02 2 gauge. One finds R =R f, which not surprisingly, is radion field is still a ghost. For a non–Fierz-Pauli mass the same as in RS1. Then starting from GNUV coordi- term on the UV brane the radion is physical but now there nates, we straighten the IR brane, keeping the UV brane is an additional state in the theory that is a ghost. straight and maintaining h5 0. In the notation of (69), It is of interest to consider whether there are simple this requires F R0 and F R0ÿ R02=R2. The com- modifications of this theory that could evade this prob- putation of the radion kinetic term proceeds as before, lem. In models of latticized gravity [29] the radion exci- and one arrives at (73). Here though one finds that the tation is absent, but unitarity is still maintained up to radion has a healthy kinetic term and is not a ghost. The scales larger than the compactification scale. It is there- radion wave function is also peaked at the IR brane and in fore conceivable that a latticized version of the model we the limit that the IR brane is decoupled the radion is not have considered could be a successful realization of a normalizable. All of these properties of the radion are theory which modifies gravity at long distances. also found to occur in the RS model. These results with a UV mass term are not surprising, since all we are doing ACKNOWLEDGMENTS here is adding a small perturbation on the UV brane where in the RS model the radion already had an expo- We would like to thank Markus Luty, Jihad Mourad, nentially small support. and Mark Wise for useful discussions. Z. C. and M. G. Finally, a minor puzzle raised in Sec. III is now re- would like to thank the hospitality of Saclay. M. G. and solved. In the computation (36) of the perturbation due to C. G. would like to thank the hospitality of Lawrence a source on the UV brane there was no term that could be Berkeley National Laboratory. C. G. and L. P. thank the interpreted as due to the exchange of a radion with non- Aspen Center for Physics for its hospitality while part of derivative couplings. The reason for this is that the non- this work was completed. The work of M. G. is supported derivative component of the radion appearing in h — by the U.S. Department of Energy under Contract
084028-11 Z. CHACKO, M. GRAESSER, C. GROJEAN, AND L. PILO PHYSICAL REVIEW D 70 084028
No. DE-FG03-92-ER40701. C. G. and L. P. are supported 2. Mass on the IR brane ( UV 0; IR 0) in part by the RTN European Program HPRN-CT-2000- In the long distance limit qR0 1, by expanding the 00148 and the ACI Jeunes Chercheurs 2068. Bessel functions around the origin we get the leading form of the propagator with both legs on the UV brane APPENDIX A: SCALAR PROPAGATOR N R; R; q2! 4 x0 ÿ x; (A7) 1. General expression ᮀ ÿ m2 The scalar Green’s function equation with mass terms with localized on the UVand IR branes and a source at z z0 1 R4 2 1 1 ÿ 04 R is solution of the bulk equation 4 R IR ; (A8) N R2 1 R2 R 1 ÿ 02 1 ÿ 2 02IRR 3 z3 R 4 R @2 ÿ @ ÿ q2 z ÿ z0; (A1) z z z 3 and R 8 R 4 supplemented by the two boundary conditions at the UV m2 IR : (A9) R R R0 and IR branes: 4 IR to leading order in R=R0. We recover the expression (58) R for the lightest graviton found in Sec. IVA. As another @zjR UVjR and @zjR0 ÿIR 0 jR0 : (A2) R check, note that in the limit IRR ! 0, the RS1 result ! 2=1 ÿ R=R02=R, is recovered: The Green’s function solution to this differential equa- N tion is obtained by first solving the homogeneous equation 2 2 1 4 0 0 0 R; R; q ! x ÿ x; (A10) to the left (z
084028-12 MASSIVE GRAVITY ON A BRANE PHYSICAL REVIEW D 70 084028 use these results in the limit that the IR brane is de- taken to be 0 coupled, 1=R ! 0. In this limit there is a mass gap, 3 Z 1 R 4 2 2 with a continuum of bulk graviton states down to zero. S d x ah ÿ bh 0 : (B2) IR 2 R0 jzR Following [25], we expect the massive graviton (A13) to 85 be unstable. To see this one has to compute the propagator The case a b ÿ2f4 R=R0 gives the Fierz-Pauli 2 2 5 IR for timelike momenta p ÿq < 0. mass term studied in Sec. II. Since this brane action is 0 Sending the IR brane to infinity, R=R ! 0, and impos- not coordinate invariant, we need to specify the coordi- ing that positive frequency waves are ingoing at z 1 nates in which the action has this form. We choose it to (or equivalently, performing the analytic continuation of describe the so-called GNIR coordinates, where h55 the propagator in (A5) and (A6)), gives h5 0 locally near the brane. 0 2 1 1 From the equations of motion we obtain the boundary zz H qz>H qz<B qz> z; z0 2 2 (A14) condition at the IR brane to be R3 1 1 qH1 qRÿUVH2 qR GNIR GNIR @zh ÿ @zh jzR0 with GNIR GNIR ah ÿ bh jzR0 : (B3) B qz qJ qRÿ J qRÿqH 1 qR > 1 UV 2 1 As in four dimensional massive gravity with a J qz ÿ H 1 qR 2 < : (A15) non–Fierz-Pauli mass term, here we expect the existence UV 2 1 H2 qz< of an additional propagating scalar degree of freedom, 1 corresponding to the trace of the metric. H J iY is the Hankel function of the first kind of Indeed, solving the bulk equations of motion and the order . boundary conditions allows for a nonzero trace of the As a check, note that in the limit of a vanishing Fierz- form Pauli mass, UV 0, we recover the RS2 propagator 1 b ÿ a found in [30]. h x; z x R02 ÿ z2 ᮀ x (B4) The interesting result is the presence of a pole at 6 a 1 where again h h and ᮀ @ @ and is a 4D scalar H qR q 1 ÿ 0: (A16) 1 UV field. The boundary condition (B3) then simply deter- H2 qR mines the mass of : This is almost identical to the equation solved by [25] in a a a ÿ 4b m2 : (B5) related context. There they found a complex pole. R0 b ÿ a Following [25], we expand this equation in the qR 1 limit using asympotic properties of the Bessel functions Next, we would like to determine whether this field and is a ghost, and whether the radion is still a ghost when the mass term is not of the Fierz-Pauli form. To this end, we 1 H qR Y1 qR J1 qR will need to compute the off-shell 4D effective action. 1 1 ÿ i (A17) 1 Y qR Y qR First note that on-shell and in GNIR coordinates the H2 qR 2 1 most general solution to the bulk equations of motion and where the ellipses denoted terms suppressed by qR.The the IR boundary condition is given by solution to (A16) is given by 4 GNIR z m m ÿ iÿ (A18) h H x; z ÿ @@f x 0 2R3 2 2 with m0 2UV=R and ÿ=m0 m0R =8. 1 z@@ x2 x (B6)
where the function 1 and the two constants 2 and are APPENDIX B: NON–FIERZ-PAULI MASS TERM given by ON THE IR BRANE R04 2 R03 This appendix analyzes the gravitational spectrum for ÿ ; (B7) 2R3 a R3 the case of a generic non–Fierz-Pauli mass term for the graviton on the IR brane. a ÿ b The bulk action is 2 ; (B8) 3a Z 5 p R S d x g (B1) 1 2 02 2 bulk 2 4b ÿ a ÿ z ÿ R b ÿ am 25 z 2 : (B9) 1 3am2 where the ... includes, in particular, the Gibbons- Hawking boundary terms. The action on the IR brane is Satisfying the boundary conditions and equations of mo-
084028-13 Z. CHACKO, M. GRAESSER, C. GROJEAN, AND L. PILO PHYSICAL REVIEW D 70 084028 tion implies that (i) H is transverse and traceless, (ii) f to the GNIR boundary conditions (B3) that were previ- is massless and it is identified with the radion of the ously inferred from the equations of motion. previous sections, and (iii) is the additional degree of As previously mentioned, to this action must be added freedom identified above with mass given by (B5). the non–Fierz-Pauli brane action. The only important To find the effective four dimensional action for these point to note is that it must be evaluated in GNIR coor- states we need to provide an off-shell decomposition of dinates. (We could evaluate it in rad coordinates, but that GNIR the metric fluctuation h. The decomposition (B6) is would involve a lengthy substitution of h in terms of unique once f is related to the brane bending of the UV hrad into the brane action.) brane in the GNIR coordinates and once is defined as After a lengthy computation, substituting (B11) into the trace of the metric fluctuation at the boundary : the bulk action (B13), using (B6), and including the brane mass term action (B2), gives (without of course using the GNIR four dimensional equations of motion) h jzR0 : (B10)
Z R0 3 3 This provides for an off-shell definition of the trace of 1 5 R a R Seff ÿ d z H EH H. 2 3 2 03 45 R z 85 R The effective action is most easily computed in the Z Z 3 4 2 2 1 4 R coordinate system where the branes are parallel and fixed d x H ÿ H 0 ÿ d x 0 jzR 2 3 at z R and z R (so-called ‘‘rad’’ coordinates). Thus 85 z in the action given below, the metric appearing there is in Z 0 3 0 0 R 4 ᮀ the rad coordinates. The metric in these coordinates is H H ÿ HH j ÿ d xf f R 22R obtained by transforming from GNIR coordinates to a 5 Z b ÿ a2R3 b ÿ a 4b ÿ aR3 coordinate system with both branes parallel. This gives 4 ᮀ 2 2 02 d x 2 03 24a 5R 24a5R rad GNIR 2 h h ÿ F z x Z R d4x2: (B14) Z z 0 z 0 ÿ 2 dz F z @@ x; (B11) R R 2z h F0 x The first two lines describe the action for the massive 55 R gravitons and their (linearized) Gibbons-Hawking terms. where Note that for the massive gravitons their mass term has been written in the Fierz-Pauli form. This guarantees that R b ÿ a xf x x (B12) for these states there are 5 on-shell degrees of freedom. 6 a The last term in the second line and all the terms in the last line describe the quadratic action for the radion and is the transformation needed to straighten the UV brane. the field. The five dimensional action is given by We briefly highlight many significant cancellations Z 3 Z 3 that occured before arriving at this result. First note that 1 5 R AB 1 4 R S ÿ 2 d x h EABhCDÿ 2 d x both the radion and the field have decoupled from each 45 z 85 z Z 3 other and from all the spin-2 gravitons. This reassures us 0 3 R 0 0 0 R 4 R that at the quadratic level (B6) correctly decouples all the h h ÿ hh jR 2 d x 4 h55hjR : 85 z fields from each other. Further, all quadratic terms in- (B13) volving more than two derivatives also canceled. From the action (B14) we find that is not a ghost, in To this must be added the non-FP brane mass term (B2). contrast to what occurs in purely four dimensional mas- Each of these terms require some explanation. The varia- sive gravity with a non-FP mass term. This may not be tion of the first term gives (5)–(7), the linearized equa- surprising, since in the AdS/CFT correspondence the non- tions of the motion in the bulk . The terms in the second FP mass term on the IR brane does not correspond in the line are the linear equivalent of the Gibbons-Hawking CFT to adding a non-FP mass term, but rather to break- terms: variation of the term on the first line produces ing general coordinate invariance in the IR.8 From the terms on the boundary that are canceled by the variation action (B14), we read off that the mass of is given by of the terms appearing in the second line. All boundary 0 terms of the type O hAB are canceled this way. Terms 8 that do not cancel are of the form h . Requiring We have explicitly checked that when a non–Fierz-Pauli O mass term is added on the UV brane the scalar field is now a that they vanish gives the boundary conditions in rad ghost as it could have also been guessed from the AdS/CFT coordinates. Using (B11), one finds they are equivalent correspondence.
084028-14 MASSIVE GRAVITY ON A BRANE PHYSICAL REVIEW D 70 084028 a a ÿ 4b These results generalize our conclusion that the radion m2 ; (B15) R0 b ÿ a is a ghost when the mass term has the Fierz-Pauli form. That is, in a theory with a non–Fierz-Pauli mass term on which agrees with the previous computation using the 5D the IR brane the radion is always a ghost. equations of motion. This provides a nontrivial consis- tency check that the computation of the effective action is Note added.—While this work was being completed, a correct. work appeared [31] that proposes a long distance modifi- We find that even for the more general non–Fierz-Pauli cation of gravity based on a Lorentz violating theory. The mass term the radion is still a ghost. The value of its model makes use of a ghost that condenses. A connection kinetic term is independent of whether or not the brane between the presence of ghosts and Lorentz violations has mass term has the Fierz-Pauli form. also recently been studied in [21].
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