Cascading Gravity: Extending the Dvali-Gabadadze-Porrati Model to Higher Dimension

PHYSICAL REVIEW LETTERS week ending PRL 100, 251603 (2008) 27 JUNE 2008

Claudia de Rham,1,2 Gia Dvali,3,4 Stefan Hofmann,1,5 Justin Khoury,1 Oriol Pujola`s,4 Michele Redi,4,6 and Andrew J. Tolley1 1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada 2Department of Physics & Astronomy, McMaster University, Hamilton ON, L8S 4M1, Canada 3CERN, Theory Division, CH-1211 Geneva 23, Switzerland 4Center for Cosmology and Particle Physics, New York University, New York, New York 10003, USA 5NORDITA, Roslagstullsbacken 23, 106 91 Stockholm, Sweden 6Institut de Theórie des Pheńome`nes Physiques, EPFL, CH-1015, Lausanne, Switzerland (Received 15 January 2008; published 27 June 2008) We present a generalization of the Dvali-Gabadadze-Porrati scenario to higher codimensions which, unlike previous attempts, is free of ghost instabilities. The 4D propagator is made regular by embedding our visible 3-brane within a 4-brane, each with their own induced gravity terms, in a flat 6D bulk. The model is ghost-free if the tension on the 3-brane is larger than a certain critical value, while the induced metric remains flat. The gravitational force law ‘‘cascades’’ from a 6D behavior at the largest distances followed by a 5D and finally a 4D regime at the shortest scales.

DOI: 10.1103/PhysRevLett.100.251603 PACS numbers: 11.25.Wx, 04.50.Kd, 98.80.Cq

Z Introduction.—The present acceleration of the Universe 1 4ᮀ 3ᮀ 2ᮀ S M M zM yz is a profound mystery. While the observational data are 2 6 6 5 5 4 4 consistent with a cosmological constant (CC) of order 103 eV4, this value is in stark disagreement with parti- describing a codimension 2 kinetic term embedded into a cle physics computations. The problem is even more severe codimension 1 one in 6D. We will impose throughout the than the hierarchy problem in the Standard Model, since Letter Z2 Z2 orbifold projection identifying y !y and dynamical solutions are impossible in theories with a z !z. The model possesses the two mass scales, m5 3 2 4 3 massless 4D graviton [1]. In the same way as the perihelion M5=M4 and m6 M6=M5. precession of Mercury was explained by a modiﬁcation of In absence of the 4D kinetic term, the propagator on the Newtonian gravity, an alternative approach is to assume codimension 1 brane (4-brane) is the DGP propagator [2], 0 that the acceleration signals a breakdown of general rela- Z 1 iqyy 0 0 1 dq e

p tivity at cosmological distances. G yy 3 ; (1) M 1 2 2 2 2 2 The Dvali-Gabadadze-Porrati (DGP) model [2] provides 5 p q 2m6 p q a simple mechanism to modify gravity at large distances by where y is the coordinate orthogonal to the codimension 2 adding a localized graviton kinetic term on a codimension brane (3-brane), p the 4D momentum, and q the momen- 1 brane in a flat 5D spacetime. The extension to higher tum along y. To find the exact 5D propagator, we can treat dimensions is particularly important both for its possible the 4D kinetic term (located at y 0) as a perturbation and embedding into string theory and for its relevance to the then sum the series. One finds, 0 0 0 2 0 2 0 0 CC problem [3,4]. However, the natural generalization of Gy; y G y y M4G yp G y the DGP model with higher codimension branes is not 4 0 4 0 0 0 straightforward [5,6]. On the one hand, these models re- M4G yp G 0G y ... quire some regularization due to the divergent behavior of M2p2 G0y y0 4 G0yG0y0: the Green’s functions in higher codimension. More seri- 1 M2p2G00 ously, most constructions are plagued by ghost instabilities 4 around flat space (not to be confused with those of the self- (2) accelerating branch of standard 5D DGP) [5,6]—see [7] In particular, the 4D brane-to-brane propagator is deter- for related work. The purpose of this Letter is to show that mined in terms of the higher dimensional Green’s function, 0 2 0 2 both pathologies can be resolved by embedding the codi- G4 G0; 0G 0=M4G 0p 1. mension 2 DGP model into a codimension 1 brane with its For the case at hand, own kinetic term. It will be interesting to see if this setup s 2 2m p allows for higher-codimension self-accelerated solutions. 0 1 6 G 0 q tanh : (3) Our present focus, however, is to derive a consistent frame- 3 2 2 2m6 p M5 4m6 p work in which gravity is modified in the infrared. Scalar.—We shall focus on the codimension 2 case. As a For p>2m6, the analytic continuation of this expression is warm-up, we consider a real scalar field with action, understood.

2 M 0 Remarkably, the 5D kinetic term makes the 4D propa- and p5 pMp . G is obtained by integrating the 5D gator finite, thereby regularizing the logarithmic diver- propagator with respect to the extra momentum. To com- gence characteristic of pure codimension 2 branes. In pute the propagator on the 3-brane, we determine the particular, when M5 goes to zero, one has G4 ! coefficients a, b, c, d, e, f through the system of linear 4 M6 logp=m6, reproducing the codimension 2 Green’s equations (5). One finds, function with a physical cutoff given by m6. The corresponding 4D Newtonian potential scales as 1=r3 at the 1 a c=I1 2 largest distances, showing that the theory becomes six 2I1p 1 dimensional, and reduces to the usual 1=r on the shortest 2 2 I1p scales. Its behavior at intermediate distances, however, b p d 2 2 I1p 1I1p 2 depends on m5;6.Ifm5 >m6, there is an intermediate 5D (7) 2 regime; otherwise, the potential directly turns 6D at a 1 4I1 3I2p 1=2 e 2 2 distance of order m5m6 logm5=m6. 3I1p 1 3I1p 2 Gravity.—Let us now turn to gravity. In analogy with the 2 2 I2 2I1 I1I2p scalar field, we consider the action, f 2 2 I1p 1I1p 2 4 Z 3 Z 2 Z M6 p M5 p M4 p 2 0 0 S g R g R g R where I M G 0=2 with G 0 defined in (3) and 2 6 6 2 5 5 2 4 4 1 4 Z 1 1 dq where each term represents the Ricci scalar. This guaran- I2 tees that the model is fully 6D general covariant. 2m6m5 1 2 To find the propagator, it is convenient to follow the 1 p p : same procedure as for the scalar and sum the diagrams with p2 q2p2 q2 2m p2 q2 insertion of the lower dimensional kinetic term, i.e., the 6 Einstein tensor E. For our purpose, we only compute the All these coefficients are finite, showing that the regulari- propagator on the 3-brane. Given the higher dimensional zation is also effective for the spin 2 case. propagator, the brane-to-brane propagator due to the in- Having determined the coefficients of the tensor H, the sertion of a codimension 1 term is, in compact form, full propagator is given by Eq. (4). To linear order the amplitude between two conserved sources on the brane is 0 2 0 1 0 G fG 1 M4EG g GH; (4) rather simple, where the first equality refers to the same resummation as 1 I I p2 1 0 1 0 1 0 T T TT ; (8) in (2) and G is the 4D part of the higher dimensional 2 2 2 M4 I1p 1 2I1p 4 Green’s function evaluated at zero. The tensor H satisfies by definition, and only depends on the first integral I1. The coefficient in front of the amplitude is exactly as for 1 2 0 the scalar; however, there is a nontrivial tensor structure. 1 M4EG H : (5) 2 One worrisome feature of this amplitude is that the relative 0 0 To find H, one can write the most general Lorentz cova- coefficient of TT and TT interpolates between 1=4 riant structure compatible with the symmetries, in the IR and 1=2 in the UV. The 1=4 in the IR gives the correct tensor structure of gravity in 6D and is unavoidable H a b cp p because at large distances, the physics is dominated by 6D Einstein term. From the 4D point of view, this can be pp pp pp dpp understood as the exchange of massive gravitons and an e pp fp p pp; extra-scalar. The 1=2 in the UV, on the other hand, signals the presence of a ghost. This agrees with previous where represents the flat Minkowski metric. results [5,6] which used a different regularization. From Requiring that this satisfies Eq. (5) leads to a system of the 4D point of view, the theory decomposes into massive linear equations whose solution determines the coefficients spin 2 fields and scalars. Since the massive spin 2 gives an a, b, c, d, e, f. Using this information, one then recon- amplitude with relative coefficient 1=3, the extra repul- structs the exact propagator from Eq. (4). sion must be provided by a scalar with wrong sign kinetic It is straightforward to apply this technique to cascading term. Separating from Eq. (8) the massive spin two con- DGP. Starting from 6D, the propagator on the 4-brane is [4] tribution, we identify the scalar propagator as

1 ~MP~NQ ~MQ~NP 2 ~MN~PQ 1 I1

G : (9) GMNPQ 3 2 (6) ghost 2 2 2M5p5 2m6p5 6M4 I1p 2 where ~ pMpN , and M; N ... are 5D indices This propagator has a pole with a negative residue; there- MN MN 2m6p5 251603-2 PHYSICAL REVIEW LETTERS week ending PRL 100, 251603 (2008) 27 JUNE 2008 ~ ~ fore, it contains a localized (tachyonic) ghost mode in that of h, and only a cross term between and h addition to a continuum of healthy modes. remains. Ghost free theory.—To clarify the origin of the ghost, it From this, it is straightforward to show the presence of a is illuminating to consider the decoupling limit studied in ghost. The scalar longitudinal component of h acquires a [8]. This will allow us to show how a healthy theory can be positive kinetic term by mixing with the graviton [8,10]. obtained by simply introducing tension on the 4D brane By taking while retaining a flat intrinsic geometry. @ @ In the 6D case, the decoupling limit [8] corresponds ~ ^ h h p ; (14) 2 3=2 2=7 ᮀ to taking M5, M6 !1 with s m6M5 m5 4 M16=M91=7 finite. In this limit, the physics on the 4-brane 6 5 one finds that there are in fact two 4D scalar modes whose admits a local 5D description, where only the nonlineari- kinetic matrix in the UV is ties in the helicity 0 part of the metric are kept, and are suppressed by the scale s. The effective 5D Lagrangian is 3 2ᮀ 11 given by M 4 (15) 4 4 10

M3 3 9 which has obviously a negative eigenvalue corresponding L 5 hMNEh M3@2 1 ᮀ 5 8 MN 2 5 32m2 5 to a ghost. 6 2 Having understood the origin of the ghost, we are now M4 ~ ~ 1 ~ z h Eh h T ; (10) ready to show how to cure it. To achieve this, we clearly 8 2 need to introduce a positive localized kinetic term for . This can arise from extrinsic curvature contributions. The where Eh ᮀh ... is the linearized Einstein MN MN simplest and most natural choice is to put a tension on tensor, M; N ... are 5D indices, and ; ... are four di- the 3-brane. This produces extrinsic curvature while leav- mensional. We have rescaled and h so that they are ing the metric on the brane flat since the tension only dimensionless, and the physical 5D metric perturbation is creates a deficit angle. ~ The solution to the 5D equations following from (10) for h MN hMN MN: (11) a 3-brane with tension is [11] The first line of (10) is the 5D version of the ‘‘ 0 0 jzj;h jzj : (16) Lagrangian’’ introduced in [8] for the DGP model. In 6M3 6M3 addition to this, we have the localized curvature term on 5 5 ~ the 3-brane, which depends on 4D physical metric h. This is an exact solution including the nonlinear terms for This introduces a kinetic mixing between and the 5D —they vanish identically for this profile. The back- metric. ground corresponds to a locally flat 6D bulk with deficit 4 We now take a further step and compute the boundary angle =M6 and with flat 4D sections. effective action valid on the 3-brane. At the quadratic The crucial point is that on this background, the order, by integrating out the fifth dimension, thep 5D kinetic Lagrangian acquires contributions from the nonlinear 3 ᮀ term of produces a 4D ‘‘mass term’’ M5 4 while terms. These can be found considering the perturbations, the Einstein tensor gives rise to Pauli-Fierz (PF) structure 0 zz; x ; for h on the boundary [9], 0 h h zh z; x ; (17) M3 p p 5 ᮀ 3 ᮀ L4 h 4h h3M 4 4 5 T T: 2 M4 ~ ~ 1 ~ h Eh h T; (12) Plugging (17)in(10) and dropping , one obtains at 8 2 quadratic order (up to a total derivative), where h and now denote the 5D fields evaluated at the 9 2 L z@ : (18) 3-brane location. 5 8 m2 In terms of the physical metric, (12) takes the form, 6

This is a localized kinetic term for that contributes to the M3 p 3 p 5 ~ ᮀ ~ ~ 3 ᮀ ~ 4D effective action with a healthy sign when > 0. L4 h 4h h M 4h 4 2 5 Therefore, for large enough , the kinetic matrix for the M2 1 2 scalars (15) becomes positive, and the ghost is absent. 4 h~Eh~ h~T : (13) 8 2 This can also be seen by computing the one particle exchange amplitude. With the addition of (18), the effec- Note that the kinetic term for is completely absorbed by tive 4D equations are 251603-3 PHYSICAL REVIEW LETTERS week ending PRL 100, 251603 (2008) 27 JUNE 2008 p p 2 ~ 3 ᮀ ~ ~ 3 ᮀ M4Eh 2M5 4h h2T 6M5 4; (19)

3 p terms. The model interpolates between a 6D behavior at ᮀ 3 ᮀ ~ M h: (20) 2m2 4 5 4 large distances and a 4D one at short distances with an 6 intermediate 5D regime. The kinetic terms regularize the Using the Bianchi identities and the conservation of T, divergent codimension 2 behavior. The model is ghost-free the double divergence of (19) leads to at least for a certain range of parameters if on the codi- p 3 ᮀ ~ ᮀ mension 2 brane there is a large enough tension. M h 6 0; (21) 5 4 E 4 We have left several questions for future study. At linear- ~ ~ ᮀ ~ where we have used Eh 2@ @ h 4h. On the level, the tensor structure of the graviton propagator is other hand, the trace of (19), in conjunction with (20) and inconsistent with observations. In the context of DGP, (21), leads to this was shown not to be a problem because the nonlinearities restore the correct tensor structure [12]. A hint of a 2 M4O 2T ; (22) similar phenomenon in the present model is given by the p 2 2 ᮀ ᮀ longitudinal terms of the graviton propagator (4). These are where O 9=m6M46 4 24m5 4. Combining (19), (21), and (22), one derives that the singular when the mass parameters m5;6 vanish and give physical metric is, up to pure gauge terms, large contributions to nonlinear diagrams. In fact, we ex- pect a ‘‘double’’ Vainshtein effect. For dense enough ~ 2 1 T 1 sources, the nonlinearities should ﬁrst decouple the extra h 2 T T ; (23) M4 O 3 O 5D scalar mode restoring 5D behavior followed by another p step to 4D. Another important direction to study is cos- where ᮀ 2m ᮀ . The tensor structure of the O 4 5 4 mology. The model has the intriguing codimension 2 fea- amplitude interpolates between 1=4 in the IR and ture that tension does not curve the space. This is obviously 1 1 3 1 of interest for the cosmological constant problem.

1 2 2 We thank Gregory Gabadadze for useful discussions. 3 6 2 M4m6 This work is supported in part by NSF Grant No. PHY- in the UV. The amplitude above corresponds to the ex- 0245068 and by the David and Lucile Packard Foundation change of massive spin 2 ﬁelds and a scalar obeying (G. D. and M. R.), by DURSI under Grant No. 2005 BP-A Eq. (22). The (DGP-like) kinetic term for the scalar is 10131 (O. P.), and by NSERC and MRI (J. K., A. J. T., positive as long as C. d. R., and S. H.).

2 2 2 > M m : (24) 3 4 6 In this regime, we see that the localized ghost disappears [1] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). and the scalar sector is composed of a healthy resonance. [2] G. R. Dvali, G. Gabadadze, and M. Porrati, Phys. Lett. B In the limit we are considering, the tension required is 484, 112 (2000). consistent with having six noncompact dimensions. [3] G. Dvali, G. Gabadadze, and M. Shifman, Phys. Rev. D Indeed, requiring that the deﬁcit angle in the bulk be less 67, 044020 (2003). than 2 leads to < 2M4. It follows that [4] G. R. Dvali and G. Gabadadze, Phys. Rev. D 63, 065007 6 (2001). 6 2 4 3M5 >M4M6; (25) [5] S. L. Dubovsky and V.A. Rubakov, Phys. Rev. D 67, 104014 (2003). which is always satisﬁed in the 5D decoupling limit and [6] G. Gabadadze and M. Shifman, Phys. Rev. D 69, 124032 displays the necessity of the induced term on the codimen- (2004). sion 1 brane. Moreover, the condition above is equivalent [7] N. Kaloper and D. Kiley, J. High Energy Phys. 05 (2007) to having m6