Negative Tension Brane Research Articles

Colliding branes in heterotic M theory

We study the collision of two flat, parallel end-of-the-world branes in heterotic M theory. By insisting that there is no divergence in the Riemann curvature as the collision approaches, we are able to single out a unique solution possessing the local geometry of (2d compactified Milne)/${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{R}}_{3}$, times a finite-volume Calabi-Yau manifold in the vicinity of the collision. At a finite time before and after the collision, a second type of singularity appears momentarily on the negative-tension brane, representing its bouncing off a zero of the bulk warp factor. We find this singularity to be remarkably mild and easily regularized. The various different cosmological solutions to heterotic M theory previously found by other authors are shown to merely represent different portions of a unique flat cosmological solution to heterotic M theory.

Gauge fields, fermions and mass gaps in 6D brane worlds

We study fluctuations about axisymmetric warped brane solutions in 6D minimal gauged supergravity. Much of our analysis is general and could be applied to other scenarios. We focus on bulk sectors that could give rise to Standard Model gauge fields and charged matter. We reduce the dynamics to Schrödinger type equations plus physical boundary conditions, and obtain exact solutions for the Kaluza–Klein wave functions and discrete mass spectra. The power-law warping, as opposed to exponential in 5D, means that zero mode wave functions can be peaked on negative tension branes, but only at the price of localizing the whole Kaluza–Klein tower there. However, remarkably, the codimension two defects allow the Kaluza–Klein mass gap to remain finite even in the infinite volume limit. In principle, not only gravity, but Standard Model fields could ‘feel’ the extent of large extra dimensions, and still be described by an effective 4D theory.

Linearized gravity in the Randall-Sundrum model with stabilized distance between branes

We consider linearized gravity in the Randall-Sundrum model in which the distance between branes is stabilized by introducing the scalar Goldberger-Wise field. We construct the second variation Lagrangian for fluctuations of gravitational and scalar fields over the background solution and investigate its gauge invariance. We obtain, separate, and solve the corresponding equations of motion. For physical degrees of freedom, we obtain the effective four-dimensional Lagrangian describing the massless graviton, massive gravitons, and the set of massive scalar fields. We also find masses and coupling constants of these fields to the matter on the negative-tension brane.

PHYSICAL DEGREES OF FREEDOM IN STABILIZED BRANE WORLD MODELS

We consider brane world models with interbrane separation stabilized by the Goldberger–Wise scalar field. For arbitrary background, or vacuum configurations of the gravitational and scalar fields in such models, we construct the second variation Lagrangian, study its gauge invariance, find the corresponding equations of motion and decouple them in a suitable gauge. We also derive an effective four-dimensional Lagrangian for such models, which describes the massless graviton, a tower of massive gravitons and a tower of massive scalars. It is shown that for a special choice of the background solution the masses of the graviton excitations may be of the order of 1 TeV, the radion mass of the order of 100 GeV, the inverse size of the extra dimension being 1 TeV. In this case the coupling of the radion to matter on the negative tension brane is approximately ten times weaker than in the unstabilized model with the same values of the fundamental five-dimensional energy scale and the interbrane distance.

Observational constraints on braneworld geometry

The low energy regime of 5D braneworld models with a bulk scalar field is studied. The setup is rather general and includes the Randall-Sundrum and dilatonic braneworlds models as particular cases. We discuss the cosmological evolution of the system and conclude that, in a two brane system, the negative tension brane is generally expected to evolve towards a null warp-factor state. This implies, for late time cosmology, that both branes end up interacting weakly. We also analyze the observational constraints imposed by solar-system and binary-pulsar tests on the braneworld configuration. This is done by considering the small deviations produced by the branes on the 4D gravitational interaction between bodies in the same brane. Using these constraints we show that the geometry around the braneworld is strongly warped, and that both branes must be far apart.

4D GRAVITY ON A BRANE FROM BULK HIGHER-CURVATURE TERMS

We study a gravity model where a tensionful codimension-one three-brane is embedded in a bulk with infinite transverse length. We find that 4D gravity is induced on the brane already at the classical level if we include higher-curvature (Gauss–Bonnet) terms in the bulk. Consistency conditions appear to require a negative brane tension as well as a negative coupling for the higher-curvature terms.

Hořava–Witten stability: eppur si muove

We construct exact time-dependent solutions of the supergravity equations of motion in which two initially non-singular branes, one with positive and the other with negative tension, move together and annihilate each other in an all-enveloping spacetime singularity. Among our solutions are the Hořava–Witten solution of heterotic M-theory and a Randall–Sundrum I type solution, both of which are supersymmetric, i.e. BPS, in the time-independent case. In the absence of branes our solutions are of Kasner type, and the source of instability may ascribed to a failure to stabilise some of the modulus fields of the compactification. It also raises questions about the viability of models based on some sorts of negative tension brane.

A non-standard matter distribution in the RS1 model and the coupling constant of the radion

In the zero-mode approximation we solve exactly the equations of motion for linearized gravity in the Randall–Sundrum model with a non-standard distribution of matter in the neighbourhood of the negative tension brane. It is shown that the form of this distribution can strongly affect the coupling of the radion to matter. We believe that such a situation can arise in models with a realistic mechanism of matter localization.

On the stability of field-theoretical regularizations of negative tension branes

Any attempt to regularize a negative tension brane through a bulk scalar requires that this field is a ghost. One can try to improve in this aspect in a number of ways. For instance, it has been suggested to employ a field whose kinetic term is not sign definite, in the hope that the background may be overall stable. We show that this is not the case; the physical perturbations (gravity included) of the system do not extend across the zeros of the kinetic term; hence, all the modes are entirely localized either where the kinetic term is positive, or where it is negative; this second type of modes are ghosts. We show that this conclusion does not depend on the specific choice for the kinetic and potential functions for the bulk scalar.

Stability of Hořava–Witten spacetimes

Hořava–Witten spacetimes necessarily include two branes of opposite tension. If these branes are BPS we are led to a puzzle: a negative tension brane should be unstable as it can lose energy by expanding, whereas a BPS brane should be stable as it resides at a minimum of the energy. We provide a detailed analysis of the energy of such braneworld spacetimes in five dimensions. This allows us to show by a non-perturbative positive energy theorem that Hořava–Witten spacetimes are stable, essentially because the dynamics of the branes is entirely accounted for by the behaviour of the bulk fields. We also perform an ADM perturbative Hamiltonian analysis at quadratic order in order to illustrate the stability properties more explicitly.

Energy scales in a stabilized brane world

Brane world gravity looks different for observers on positive and negative tension branes. First we consider the well-known RS1 model with two branes embedded into the AdS 5 space–time and recall the results on the relations between the energy scales for an observer on the negative tension brane, which is supposed to be “our” brane. Then from the point of view of this observer we study energy scales and masses for the radion and graviton excitations in a stabilized brane world model. We argue that there may be several possibilities leading to scales of the order 1–10 TeV or even less for new physics effects on our brane. In particular, an interesting scenario can arise in the case of a “symmetric” brane world with a nontrivial warp factor in the bulk, which however takes equal values on both branes.

Conformal symmetry of brane world effective actions

A simple derivation of the low-energy effective action for brane worlds is given, highlighting the role of conformal invariance. We show how to improve the effective action for a positive- and negative-tension brane pair using the AdS/CFT correspondence.

Cosmological perturbations during radion stabilization

We consider the evolution of cosmological perturbations during radion stabilization, which we assume to happen after a period of inflation in the early universe. Concentrating on the Randall-Sundrum brane world scenario, we find that, if matter is present both on the positive and negative tension branes, the coupling of the radion to matter fields could have significant impact on the evolution of the curvature perturbation and on the production of entropy perturbations. We investigate both the case of a long-lived and a short-lived radion and outline similarities and differences to the curvaton scenario.

Effective actions of a Gauss–Bonnet brane world with brane curvature terms

We consider a warped brane world scenario with two branes, Gauss–Bonnet gravity in the bulk, and brane localised curvature terms. When matter is present on both branes, we investigate the linear equations of motion and distinguish three regimes. At very high energy and for an observer on the positive tension brane, gravity is four-dimensional and coupled to the brane bending mode in a Brans–Dicke fashion. The coupling to matter and brane bending on the negative tension brane is exponentially suppressed. In an intermediate regime, gravity appears to be five dimensional while the brane bending mode remains four-dimensional. At low energy, matter on both branes couple to gravity for an observer on the positive tension brane, with a Brans–Dicke description similar to the 2-brane Randall–Sundrum setup. We also consider the zero mode truncation at low energy and show that the moduli approximation fails to reproduce the low energy action.

Gauss-Bonnet gravity renders negative tension braneworlds unstable

We show that the addition of the Gauss-Bonnet term to Einstein gravity induces a tachyon mode in the spin two fluctuations of the Randall-Sundrum I model. We demonstrate that this instability is generically related to the presence of a flat negative tension brane, of codimension one, embedded in an anti-de Sitter background. In particular its presence is independent of ${Z}_{2}$-symmetry or compactness of the extra dimension. The gravitational tachyon mode persists for arbitrarily small but nonvanishing Gauss-Bonnet coupling. It is a bound state localized on the negative tension brane, much like the graviton zero-mode is localized on a positive tension one. We discuss the possible resolution of this instability by the inclusion of induced gravity terms on the branes or by an effective four-dimensional cosmological constant.

Randall-Sundrum two D-brane model

In Randall-Sundrum two D-brane system we derive the gravitational theory on the branes. It is turned out from the consistency that one D-brane has the negative tension brane under Randall-Sundrum tuning and both gauge fields on the brane are related by scale transformation through the bulk RR/NS-NS fields. As same with the single D-brane case, the gauge field which is supposed to be localised on the brane does not couple to the gravity on the branes.

Boundary inflation in the moduli space approximation

The evolution of slow--roll inflation in a five--dimensional brane world model with two boundary branes and bulk scalar field is studied. Assuming that the inflationary scale is below the brane tension, we can employ the moduli space approximation to study the dynamics of the system. Detuning the brane tension results in a potential for the moduli fields which we show will not support a period of slow--roll inflation. We then study an inflaton field, confined to the positive tension brane, to which the moduli fields are non--minimally coupled. We discuss in detail the two cases of $V(\chi) = {1/2} m^2 \chi^2$ and $V(\chi) = \lambda \chi^4$ and demonstrate that increasing the coupling results in spectra which are further away from scale--invariance and in an increase in the tensor mode production, while entropy perturbations are subdominant. Finally, we point out that the five--dimensional spacetime is unstable during inflation because the negative tension brane collapses.

Solutions of multigravity theories and discretized braneworlds

We determine solutions to 5D Einstein gravity with a discrete fifth dimension. The properties of the solutions depend on the discretization scheme we use and some of them have no continuum counterpart. In particular, we find that the negligence of the lapse field (along the discretized direction) gives rise to a Randall–Sundrum-type metric with a negative tension brane. However, no brane source is required. We show that this result is robust under changes in the discretization scheme. The inclusion of the lapse field gives rise to solutions whose continuum limit is gauge fixed by the discretization scheme. We find, however, one particular scheme which leads to an undetermined lapse reflecting the reparametrization invariance of the continuum theory. We also find other solutions, with no continuum counterpart with changes in the metric signature or avoidance of singularity. We show that the models allow a continuous mass spectrum for the gravitons with an effective 4D interaction at small scales. We also discuss some cosmological solutions.

Some Properties of Multigravity Theories and Discretized Brane Worlds

We review some properties of solutions to 5D Einstein gravity with a discrete fifth dimension. Those properties depend on the discretization scheme we use. In particular, we find that the neglect of the lapse field (along the discretized direction) gives rise to Randall–Sundrum-type metric with a negative tension brane. However, no brane source is required. The inclusion of the lapse field gives rise to solutions whose continuum limit is gauge fixed by the discretization scheme. We show also that the models allow a continuous mass spectrum for the gravitons with an effective 4D interaction at small scales.

Moduli stabilization in higher dimensional brane models

We consider a class of warped higher dimensional brane models with topology × Σ × S1/2, where Σ is a D2 dimensional manifold. Two branes of codimension one are embedded in such a bulk space-time and sit at the orbifold fixed points. We concentrate on the case where an exponential warp factor (depending on the distance along the orbifold) accompanies the Minkowski and the internal space Σ line elements. We evaluate the moduli effective potential induced by bulk scalar fields in these models, and we show that generically this can stabilize the size of the extra dimensions. As an application, we consider a scenario where supersymmetry is broken not far below the cutoff scale, and the hierarchy between the electroweak and the effective Planck scales is generated by a combination of redshift and large volume effects. The latter is efficient due to the shrinking of Σ at the negative tension brane, where matter is placed. In this case, we find that the effective potential can stabilize the size of the extra dimensions (and the hierarchy) without fine tuning, provided that the internal space Σ is flat.