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

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Cascading Gravity: Extending the Dvali-Gabadadze-Porrati Model to Higher Dimension PHYSICAL REVIEW LETTERS week ending PRL 100, 251603 (2008) 27 JUNE 2008 Cascading Gravity: Extending the Dvali-Gabadadze-Porrati Model to Higher Dimension 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 The´orie des Phe´nome`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 y z is a profound mystery. While the observational data are 2 6 6 5 5 4 4 consistent with a cosmological constant (CC) of order 10ÿ3 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 iq yÿy 0 0 1 dq e p tivity at cosmological distances. G yÿy 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 G y; y G y ÿ y ÿM4G yp G ÿy 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 ÿy ... quire some regularization due to the divergent behavior of M2p2 G0 y ÿ y0ÿ 4 G0 yG0 ÿy0: the Green’s functions in higher codimension. More seri- 1 M2p2G0 0 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 G 0; 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 ÿ p allows for higher-codimension self-accelerated solutions. 0 ÿ1 6 G 0 q tanh : (3) Our present focus, however, is to derive a consistent frame- 3 2 2 2m6 p M5 4m6 ÿ 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. 0031-9007=08=100(25)=251603(4) 251603-1 © 2008 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 100, 251603 (2008) 27 JUNE 2008 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 ÿ4 M6 log p=m6, reproducing the codimension 2 Green’s equations (5). One finds, function with a physical cutoff given by m6. The corre- sponding 4D Newtonian potential scales as 1=r3 at the 1 a c=I1 ÿ 2 largest distances, showing that the theory becomes six 2 I1p 1 dimensional, and reduces to the usual 1=r on the shortest 2 2 I1p scales. Its behavior at intermediate distances, however, b ÿp d 2 2 I1p 1 I1p ÿ 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 ÿ1=2 e 2 ÿ 2 distance of order m5m6 log m5=m6. 3 I1p 1 3 I1p ÿ 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 1 I1p ÿ 2 4 Z 3 Z 2 Z M6 p M5 p M4 p 2 0 0 S ÿg R ÿg R ÿg 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 q2 p2 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 ÿ1 0 G fG 1 ÿ M4EG g G H ; (4) rather simple, where the first equality refers to the same resummation as 1 I I p2 ÿ 1 0 1 0 1 0 ÿ T T ÿ TT ; (8) in (2) and G is the 4D part of the higher dimensional 2 2 2 M4 I1p 1 2I1p ÿ 4 Green’s function evaluated at zero. The tensor H satis- fies 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 ÿ 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 ÿ1=4 riant structure compatible with the symmetries, in the IR and ÿ1=2 in the UV. The ÿ1=4 in the IR gives the correct tensor structure of gravity in 6D and is unavoidable H a b c p p because at large distances, the physics is dominated by 6D Einstein term. From the 4D point of view, this can be pp p p pp dp p understood as the exchange of massive gravitons and an e p p fp p p p ; extra-scalar. The ÿ1=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.
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