Brane Induced Gravity, its Ghost and the Cosmological Constant Problem

Brane Induced Gravity, its Ghost and the Cosmological Constant Problem
Preprint typeset in JHEP style - HYPER VERSION
arXiv:1007.1263v2 [hep-th] 26 May 2011
Brane Induced Gravity, its Ghost and the
Cosmological Constant Problem
S. F. Hassan
Department of Physics & The Oskar Klein Centre for Cosmoparticle Physics,
Stockholm University, AlbaNova University Centre, SE-106 91 Stockholm, Sweden
E-mail: [email protected]
Stefan Hofmann
Arnold Sommerfeld Center for Theoretical Physics, LMU,
Theresienstr. 37, 80333 Munich, Germany , &
Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany
E-mail: [email protected]
Mikael von Strauss
Department of Physics & The Oskar Klein Centre for Cosmoparticle Physics,
Stockholm University, AlbaNova University Centre, SE-106 91 Stockholm, Sweden
E-mail: [email protected]
Abstract: “Brane Induced Gravity” is regarded as a promising framework for addressing
the cosmological constant problem, but it also suffers from a ghost instability for parameter values that make it phenomenologically viable. We carry out a detailed analysis of
codimension > 2 models employing gauge invariant variables in a flat background approximation. It is argued that using instead a curved background sourced by the brane would
not resolve the ghost issue, unless a very specific condition is satisfied (if satisfiable at all).
As for other properties of the model, from an explicit analysis of the 4-dimensional graviton
propagator we extract a mass, a decay width and a momentum dependent modification of
the gravitational coupling for the spin 2 mode. In the flat space approximation, the mass of
the problematic spin 0 ghost is instrumental in filtering out a brane cosmological constant.
The mass replaces a background curvature that would have had the same function. The
optical theorem is used to demonstrate the suppression of graviton leakage into the uncompactified bulk. Then, we derive the 4-dimensional effective action for gravity and show
that general covariance is spontaneously broken by the bulk-brane setup. This provides a
natural realization of the gravitational Higgs mechanism. We also show that the addition
of extrinsic curvature dependent terms has no bearing on linearized brane gravity.
Keywords: Modified gravity, Cosmological applications of theories with
dimensions.
extra
Contents
1. Introduction and Results
1.1 An overview of BIG, its promise and problems
1.2 Overview and discussion of our results
1.3 Organization of the paper
2
2
4
7
2. Preliminaries
2.1 The model and its origins
2.2 Flat background and gauge invariant variables
2.3 Thick branes and “blurred” quantities
2.4 Bulk-brane relations
8
8
9
10
11
3. The
3.1
3.2
3.3
12
12
15
16
Basic Brane Induced Gravity Model
Equation of motion and solutions
Graviton decay from optical theorem
Graviton mass, decay width and coupling
4. Screening of Λ and the Origin of Tachyon/Ghost Problem
4.1 Filtering out of brane cosmological constant
4.2 On the origin of the tachyon and ghost problems
4.3 Implication for couplings
4.4 Contrast with Fierz-Pauli Massive gravity
20
21
23
24
25
5. Inclusion of Extrinsic Curvature Terms
5.1 Extrinsic curvature term in the action
5.2 Equations of motion with extrinsic curvature contributions
5.3 Solutions with extrinsic curvature contributions
25
26
26
27
6. Revisiting Fierz-Pauli Massive Gravity
6.1 Fierz-Pauli massive gravity in gauge invariant variables
6.2 A 1-parameter family of actions for FP gravity
29
29
31
7. 4-d
7.1
7.2
7.3
32
32
34
36
Effective Action in BIG and Broken Gauge Invariance
4 -dimensional effective action in Brane Induced Gravity
A 1-parameter family of 4-d BIG effective actions
Spontaneous breaking of 4-d gauge invariance
8. Conclusions
37
A. Projection operators for gauge invariant variables
38
–1–
1. Introduction and Results
“Brane induced gravity” (BIG)[1, 2] (for a review see [3]) is regarded as providing one of
the few promising frameworks for addressing the cosmological constant problem [4] in the
sense of explaining how the observed value of vacuum energy could be so small as compared
to the types of values inferred from quantum field theory [5, 6] (other approaches include
[7, 8], for reviews see [9, 10, 11]). In its basic form, the model regards the 4-dimensional
universe as a 3-brane in a non-compact (4 + n)-dimensional bulk spacetime. Gravitational
dynamics arise from Einstein-Hilbert actions both in the bulk and on the brane. Bulk
gravity absorbs the effect of brane cosmological constant, leaving behind effectively 4dimensional gravity on the brane with a “filtered out” cosmological constant. Unfortunately
consistent constructions are still lacking, for example, issues related to the presence of a
tachyonic ghost mode have not been resolved satisfactorily. Hence more work is needed to
obtain consistent working models of this type1 .
1.1 An overview of BIG, its promise and problems
Let’s briefly review the origin, workings and shortcomings of the model.
Historical background and motivation: In ordinary Einstein-Hilbert gravity a
cosmological constant Λ in a 3 + 1 dimensional spacetime gives rise to a de Sitter metric.
However, if this specetime is a 3-brane embedded in a 4 + n dimensional bulk, then it
is known that bulk gravity has classical solutions in which Λ (which is the same as the
brane tension) curves directions transverse to the brane while the metric along the 3-brane
remains flat, provided n > 1 [14, 15, 16]. This eliminates Λ from the brane point of view
at the level of classical solutions. But in this model, gravity is a (4 + n)-dimensional force.
To get effectively 4-dimensional gravity on the brane without compactifying the extra
dimensions, Dvali, Gabadadze and Porrati [1, 2] proposed adding to the bulk gravity action,
a 4-dimensional Einstein-Hilbert action for the brane induced metric. Such brane localized
actions, along with extra corrections, naturally arise in field theory [17] as well as in string
theory [18, 19, 20, 21, 22, 23] setups, hence the name brane induced gravity. However, for
phenomenological reasons, the parameters have to be tuned to make the brane localized
action dominant. The n = 1 case is the well studied DGP model [1, 3]. It is special in that
it does not suffer from tachyon/ghost problems. But it is also not adequate for addressing
the cosmological constant problem in the sense stated above. In [5, 6] it was argued that
models with n > 2 should be able to filter out a brane cosmological constant, making
gravity on the brane less sensitive to it, while at the same time giving rise to essentially 4
dimensional gravitational interactions on the brane. These arguments were mainly based
on the classical solutions of [14, 15] described above 2 .
Beside the classical solutions, most of the explicit work on BIG with n > 2 has focused, for reasons of technical simplicity, on the Λ = 0 case, analyzing the propagation of
1
In this paper we will not discuss variants of the basic setup like cascading gravity [12] that address the
ghost issue [13].
2
The n = 2 models also filter out Λ but they share features of both n = 1 and n > 2 models and are
discussed in some detail in [24, 25, 26].
–2–
linearized metric fluctuations for a flat brane, in flat bulk spacetime. The purpose is to,
from this, glean information on infrared modifications of gravity that would eventually result in filtering out Λ. In particular, one finds that for appropriate values of parameters in
the action, gravity can be made effectively 4-dimensional over a desired range of distances.
It will also contain massive unstable gravitons [2, 5, 6, 27]. Beyond this range (say, for
distances of the order of the size of the Universe), the emergent unstable graviton, with a
small but momentum dependent mass, produces infrared modifications of gravity. This is
supposed to be relevant to filtering out Λ, as suggested by the classical solutions mentioned
above.
Problematic issues: The above picture looks promising, except for some important
issues that arise in the analysis carried out so far, and that need to be better understood
and resolved. We describe five issues below.
1. The evidence in [5, 6] that BIG would filter out a brane cosmological constant is
mostly based on the existence of the classical solutions of [14, 15] that are supposed
to describe how Λ curves the space transverse to the brane. However a closer examination shows that in the vicinity of the brane, the source structure implied by
these solutions does not correspond to the brane source (which is a common problem of brane solutions). This means that a detailed analysis of fluctuations (needed
to obtain the effective 4-dimensional gravity on the brane) cannot be carried out
around existing classical solutions 3 . To avoid this problem, in this paper we start
with a flat background approximation and try to glean information about the effects
of background curvature sourced by the brane.
2. Second, as pointed out in [28], the analysis of BIG in a flat background approximation
indicates the presence of a tachyonic ghost mode and, therefore, an inconsistency of
the model. This is a major drawback and resolving the ghost problem is crucial for
developing these models further. It is easy to see that a ghost generically appears in
massive gravity theories unless it is given an infinite mass, as in Fierz-Pauli gravity
[29, 30, 31, 32, 33]. But in BIG around flat background, the tachyonic ghost cannot
be avoided easily. There have been attempts to evade the problem by modifying the
setup [13, 27, 34], but a satisfactory resolution is still lacking.
3. The third issue is the mode of realization of the gauge symmetry, in this case 4dimensional general covariance on the brane. This is related to the structure of the
effective theory on the brane. It is known that in both Fierz-Pauli massive gravity, as
well as in Higgs gravity setups (see [35, 36]), a graviton mass is related to the breaking
of 4-d general covariance. However, in BIG, the starting theory is manifestly invariant
under bulk and brane gauge transformations, while at the same time 4-d gravitons
acquire an effective mass. In usual treatments this result is obtained after gauge
fixing which obscures the nature of symmetries of the effective theory. How is the
generation of mass related to the realization of symmetry in BIG models?
3
Progress in this direction will be reported elsewhere.
–3–
4. The fourth issue is a more detailed understanding of the technical aspects of the
resulting modified gravity on the brane and its response to the cosmological constant,
both in flat and curved background approximations. Technically, a zero thickness
brane should consistently be replaced by one with an effective width [2, 17, 23],
before a quantitative investigation of graviton mass, its decay and other possible
modifications of the gravitational force. It is also important to see explicitly how a
cosmological constant affects the situation. In this paper such an analysis is carried
out in the flat background approximation.
5. Related to this is the fifth issue of naturalness of scales in BIG, at least based on
some elementary considerations. The basic BIG model contains bulk and brane
Planck scales, say, M∗ and MP , respectively, as free parameters. For a realistic
MP ∼ 1019 GeV , to insure that the resulting IR modifications of gravity are not too
large, the bulk gravity scale M∗ should be unnaturally low (< 10−3 eV for n ≥ 2),
corresponding to strongly coupled bulk gravity [2]. The effective brane thickness
introduces another scale, though often this is taken to be related to M∗ [17, 23].
But a non-zero Λ also affects the identification of the gravitational constant and
other standard model couplings, influencing the problem of scales. Taking this into
account is important since as a viable theory BIG should remain calculable at the
interacting level and should not generate new hierarchy problems. It is not obvious
that BIG could meet these requirements.
Considering the potential ramifications of brane induced gravity for the cosmological
constant problem, we revisit these models and analyze them in some depth in the hope of
gaining a better understanding of the issues outlined above. This work address some of
these issues directly, and hopefully sheds light on, and sharpens the context of, the rest.
1.2 Overview and discussion of our results
This subsection contains a description of our results and their implications. We concentrate
mainly on codimension n > 2 BIG models relevant to the cosmological constant problem
and allow for the addition of extrinsic curvature terms and a brane cosmological constant
(brane tension) Λ. In this model, we study linearized metric fluctuations around a background configuration which is approximated as a flat 3-brane in a flat bulk spacetime (for
the reason explained under item (1) of the previous subsection). The crucial point is that
the presence of a small brane cosmological constant does not invalidate the flat background
approximation.
Spontaneous breaking of general covariance: The basic variables are the bulk
metric GM N and the brane embedding functions xM (σ). The symmetries are the brane
and bulk general coordinate transformations, involving σ µ and X M , respectively. The
standard gauge choice (Monge or static gauge for the brane, and harmonic gauge for the
bulk symmetries) is natural for bulk physics but obscures the nature of symmetries of the
effective brane theory. Thus, with a focus on 4-dimensional physics, we work with gauge
invariant variables and the induced metric gµν = GM N ∂µ xM ∂ν xN , that naturally couples
–4–
to brane matter. The outcome is that, beside the usual gauge invariant parts of the metric,
we encounter 4 new gauge invariant variables. These are closely related to Stückelberg
fields used to restore gauge invariance in Fierz-Pauli massive gravity. The 4-dimensional
effective action, obtained after integrating out all extra dimensional modes, depends on
these variables, and through them, on gauge dependent components of the metric. This
explicitly demonstrates how 4-dimensional general covariance is spontaneously broken by
the bulk-brane setup in the effective 4-dimensional theory, whereas all symmetries are
manifest in the starting action. Thus, BIG naturally implement a gravitational Higgs
mechanism of the sort discussed in [35, 36]. These issues are discussed in sections 6 and 7.
Explicit analysis of graviton mass and decay: To study graviton propagation
on the brane, one has to consider thick branes to avoid divergences associated with vanishing brane width. It is then known that integrating out bulk modes contributes a “selfenergy” term to the 4-dimensional graviton propagator, giving, schematically (see equations
(3.15,3.16,2.13)),
(0,2)
Oµνρσ
.
(1.1)
Bk 2 + Aω n−2 N (0,2) Σ(k)
A and B are inverse Newton constants in d and 4 dimensions, ω is the brane width and
the superscripts refer to spin 0 and 2 modes of graviton. The scale of graviton mass is set
by Aω n−2 /B, taken to be very small for phenomenological reasons.
1. After reviewing the well known problem of the tachyonic ghost nature of the spin 0
mode, we use a Gaussian profile for the brane thickness to explicitly extract from
Σ(k) a mass term, a decay width and a momentum dependent modification of the
Newton constant for the healthy spin 2 mode. All this hinges on the fact that, as
we show, the standard perturbative treatment of unstable particles in quantum field
theory remains applicable for small graviton masses.
2. The suppression of the probability for graviton escape into the bulk, resulting in
effectively 4-dimensional gravity on the brane, is demonstrated using the optical
theorem.
Suppression of cosmological constant: In Einstein-Hilbert gravity, a cosmological
constant Λ, however small, destabilizes flat space, giving rise to de Sitter or anti de Sitter
spacetimes. In BIG, a brane Λ can curve directions transverse to the brane, leaving the
brane metric flat. However, in the analysis of BIG around a flat background, commonly
employed to extract more detailed information about brane gravity, the setup does not
allow for absorbing Λ in the curvature of extra dimensions. But, as discussed in section 4,
the theory compensates for this and neutralizes Λ using a Σ(0) 6= 0 piece of the self-energy
in the scalar mode propagator, closely related to its tachyonic mass. This leaves the brane
metric flat4 , and only modifies gravitational and non-gravitational couplings. Based on
this analysis, the following observations should be made:
4
The n > 2 BIG only attempts to explain why one does not observe a large cosmological constant. At
least in this form, it does not explain the observed recent accelerated expansion due to a very small Λ.
–5–
1. As far as filtering out Λ is concerned, the tachyonic mass of the scalar ghost replaces
the function of a background curvature (as an artifact of flat background approximation) and is not unhealthy. In fact, the mass is tachyonic only with respect to the
wrong sign of the ghost kinetic term, but has a positive contribution to the energy.
2. The momentum dependence of Σ(k), which determines the graviton decay width and
normalizes the gravitational coupling, has no bearing on the suppression of Λ (which
is entirely due to Σ(0)). This is to be contrasted with the more general degravitation
mechanisms [37] where a momentum dependent gravitational coupling could make
the theory less sensitive to Λ. But the outcome of n > 2 BIG is closer to the scenario
in [38], more closely modeled around massive gravity.
3. In some of the literature on BIG, the suppression of Λ is heuristically explained in
terms of 3-brane gravity becoming 4 + n-dimensional, hence weaker, at very large
distances. This truly is the behaviour of n = 1 theory which, however, does not
filter out Λ. A closer inspection of the propagator in n > 2 models shows that brane
gravity at large distances remains 4-dimensional and is dominated by Σ(0), related
to graviton mass, which suppresses Λ.
4. A lesson from this analysis is that the flat background approximation to BIG remains
valid in the presence of a small cosmological constant, with implications for the
resolution of the ghost problem to be discussed below.
Extrinsic curvature effects: Derivations of the BIG action in string theory setups
also produce terms dependent on the extrinsic curvature of the brane. One may wonder
if neglecting these terms has an effect on the ghost problem. The analysis in section 5
shows that these terms have no effect on brane gravity at the linearized level. This may be
disappointing from the point of view of the ghost problem but also shows that these terms
do not drastically modify gravity on the brane.
Status of the ghost problem: In equation (1.1) for the graviton propagators in
(0,2)
BIG, Oµνρσ contain the tensor structure. This is exactly the same as the tensor structure
of Einstein-Hilbert gravity in 4 dimensions (obtainable by setting A = 0). It is known that
with this tensor structure, the theory is ghost free only for zero masses. Any non-zero mass
pole implies a ghost, as is the case with BIG5 . Resolving this problem is crucial for any
eventual application of BIG. We do not achieve this in this paper, but sharpen the context
of the problem.
1. As a warm up consider the following puzzle: The BIG action can arise within seemingly consistent setups, like the string theory setup of [18]. How can these consistent
setups lead to a ghost inconsistency? To answer this, note that the identification of
ghost in BIG hinges on the Σ term in (1.1) being a small self-energy correction. This
5
The ghost can also be avoided by changing the tensor structure gravity in BIG for example, as in [27, 34]
but that introduces vDVZ type discontinuity in the free theory. These could be avoided in the non-linear
regime by the Vainshtein mechanism [39, 40], but we will not consider such alternatives here.
–6–
is obviously the case in the phenomenologically interesting regime of small graviton
mass (∼ Aω n−2 /B) which requires that the brane Einstein-Hilbert action dominates
over the bulk one. If this is not the case, then either the ghost may have a large
mass, beyond the regime of validity of the low-energy effective action, or Σ(k) may
contain appreciable corrections to Bk 2 making the above interpretation of the propagator meaningless. This is the case in the string theory regime of [18] where, for a
small string coupling gs , A ∼ 1/gs2 dominates over B ∼ 1/gs (assuming ω ∼ gs0 ), the
propagator is essentially Σ−1 (k), and gravity is d-dimensional and ghost free. On the
other hand, assuming ω 4 ∼ gs and n = 4 (more natural for D3-branes), the mass is
at string scale.
2. Could the appearance of ghost (in the phenomenologically interesting cases) be related to the use of flat background approximation, ignoring Λ and its back reaction
on bulk geometry? As discussed in section 4, since our analysis is already valid for a
small Λ, one can answer this in the negative by a continuity argument: that in any
case, the small Λ limit should correspond to our result which is not ghost free.
One may formulate a sharper criterion: In a gravitational background sourced by
a brane Λ, the metric along the brane remains flat, hence the background mostly
affects gravitational dynamics in the bulk. In terms of the propagators in (1.1), the
curved background mostly affects the A-dependent self-energy term Σ, leaving Bk2
and tensor structure unchanged. Then in the phenomenologically interesting regime
of small mass, where the B term dominates, the propagator has the same structure
as in flat background. The discussion above then shows that the ghost mode can be
avoided only if the new self-energy term satisfies Σcurved (0) = 0, implying no hard
masses. This can be regarded as the constraint on admissible ghost free backgrounds,
if any. For more on this see the concluding section.
3. In the 4-dimensional effective action for BIG derived in section 7, the ghost can
be related to the Stückelberg-like fields that arise in the gauge invariant formalism,
and hence to the spontaneous breakdown of general covariance. This underlines the
similarity with the ghost in standard massive gravity and in Higgs gravity.
1.3 Organization of the paper
The rest of the paper is organized as follows: In section 2, we introduce the model and
detail the setup for the analysis. Gauge invariant variables are introduced and a brane
thickness profile is used to avoid divergences associated with zero-thickness branes.
In section 3, we solve the equations for the basic BIG model and review the tachyonic
ghost problem. The suppression of graviton escape into bulk is described in terms of
the optical theorem. We then carry out an explicit analysis of the graviton propagator,
extracting from it a mass, a decay width and a modulation of the gravitational coupling.
It is shown in the process that standard perturbative QFT methods remain applicable for
small graviton masses.
In section 4, we consider the effect of a small brane cosmological constant Λ in the flat
background approximation, showing that for codimension n > 2, it is filtered out by the
–7–
tachyonic scalar mass. The tachyon problem is dismissed. The ghost problem is argued to
survive, even when the theory is analyzed in a background sourced by Λ, except for special
backgrounds (if they exist at all) that do not make brane gravity massive.
In section 5, we solve the equations in the presence of extrinsic curvature terms that
generically arise in derivations of the BIG action. It is shown that they have no effect
whatsoever on the brane gravity.
In section 6, we revisit massive Fierz-Pauli gravity with Stückelberg fields in terms
of gauge invariant variables. In particular, it is shown that the correct solutions can be
obtained from a 1-parameter family of actions.
In section 7, we derive an effective action for the brane gravity by integrating out
bulk related modes. We construct a 1-parameter family of effective actions that reproduce
the correct solution and, generically, contain Stückelberg fields. This shows that gauge
dependent modes of metric do not decouple, implying a spontaneous breakdown of 4dimensional general covariance. The relation to the gravitational Higgs mechanism is also
pointed out. The conclusions are summarized in section 8, and the appendix contains a
brief discussion of gauge invariant variables and related zero mode issues.
2. Preliminaries
In this section we introduce the brane induced gravity model and describe our setup for
its analysis: the use of gauge invariant variables, description of thick branes in terms of a
density profile and its use as a regulator. Finally we consider the bulk-brane relations and
the appearance of 4 new gauge invariant variables peculiar to the bulk-brane setup.
2.1 The model and its origins
In brane induced gravity models, the Universe is regarded as a 3-brane with coordinates
σ µ (µ = 0, · · · , 3) embedded in a non-compact (d = 4 + n)-dimensional bulk spacetime with
coordinates xM through xM (σ). In the basic model, dynamics of gravity is governed by an
bulk for the bulk metric G
brane for
Einstein-Hilbert action SEH
M N (x) and a similar action SEH
M
N
brane
the brane induced metric, gµν = ∂µ x ∂ν x GM N , sourced by brane matter Sm
[1, 2].
In this paper, we also allow for adding a brane tension (brane cosmological constant) term
brane , to be specified later. Then6 ,
SΛbrane and extrinsic curvature corrections SΩ
Z
Z
√
√
brane
brane
+ SΛbrane + Sm
.
(2.1)
S = −A dd x −GR(d) − B d4 σ −gR(4) + SΩ
As argued in [1, 2, 17], such models naturally arise in braneworld setups as a result of
integrating out massive matter on the brane following [42]. More specifically, they were
shown to arise in non-supersymmetric string theory, as a result of closed string scattering
brane can be computed explicitly7 .
off D-branes [18, 19] where the action (2.1), including SΩ
6
We follow the sign conventions of Weinberg in [41]
The calculation of [18, 19] leads to the expected behaviour of A ∼ gs−2 and B ∼ gs−1 with the string
coupling gs . Mass dimensions are supplied by the string tension, 1/α′ . A curious feature of their result is
brane
brane
that while, A > 0, one gets B < 0. Also, the relative sign between SEH
and SΩ
is opposite to what
appears in the Gauss-Codazzi equation. However, for the BIG results to be relevant to phenomenology, we
must take B > 0 and adjust the parameter values appropriately, as will be specified later.
7
–8–
At this order in the α′ perturbation theory, there are also bulk R2 terms, in the GaussBonnet combination, that do not contribute to our analysis, as well as terms involving
other massless string states that are ignored here. A different string theory realization was
proposed in [20]. Here for the couplings A, B, etc. we do not use their calculated high
energy values, but treat them as low energy phenomenological parameters.
The d = 5 model is the well known DGP model that has been extensively investigated.
As argued in [5, 6] it cannot address the problem of the observed smallness of cosmological
constant. d = 6 is also special and was investigated in [24, 25, 26]. Below, we are interested
mainly in d > 6 models.
2.2 Flat background and gauge invariant variables
We investigate brane induced gravity at the linearized level around a flat background,
which is technically the easiest and a good first approximation. Consider a flat bulk with
a flat brane and split {xM } into {xµ|| , xi⊥ }, respectively, parallel and perpendicular to the
brane. The flat brane corresponds to xµ|| = σ µ , xi⊥ = y0i (constants). Linearizing GM N (x),
gµν (σ) and xM (σ) around this background gives the fluctuations HM N , hµν , f µ and y i ,
GM N = ηM N + HM N (x) ,
gµν = ηµν + hµν (σ) ,
(2.2)
xµ|| (σ) = σ µ + f µ (σ) ,
xi⊥ (σ) = y0i + y i (σ) .
(2.3)
To first order, h(x(σ)) = h(x|| ), etc., and the fluctuations are related by the linearized
pullback equation hµν (x|| ) = Hµν (x|| , y0 ) + ∂µ fν + ∂ν fµ . Infinitesimal bulk and brane
diffeomorphisms transform the fluctuations as,
δxM = ξ M :
δξ HM N = −2∂(M ξN ) ,
δσ µ = λµ :
δλ HM N = 0 ,
δξ hµν = 0 ,
δξ f µ = ξ µ ,
δλ hµν = −2∂(µ λν) , δλ f µ = −λµ .
(2.4)
The last equation arises since xM (σ) are scalars under brane diffeomorphism so that δλ f µ ≡
δλ xµ = −λν δνµ . The fµ play a role similar to Stückelberg fields in massive gravity.
To solve the equations of motion, we use gauge invariant variables instead of gauge
fixing transformations (2.4)8 . This disentangles the spin 0 and spin 2 modes of the graviton
and promotes fµ to new gauge invariant variables. The bulk field HM N decomposes as
⊥
HM N = HM
N + ∂M AN + ∂N AM + ∂M ∂N Φ +
1
ηM N S
d
(2.5)
⊥
⊥M = 0, and ∂ M A
⊥
where ∂ M HM
M = 0. Then, HM N and S are gauge invariant
N = 0, HM
while AM and Φ give rise to the transformations of HM N in (2.4). The relevant projection
operators are listed in appendix A. The bulk Einstein-Hilbert action becomes,
Z
A
bulk
dd x ∂ L H M N ∂L HM N − ∂ L H∂L H + 2∂L H∂N H LN − 2∂M H M N ∂ L HLN
SEH
=−
4
Z
A
(d − 2)(d − 1) L
d
L ⊥M N
⊥
∂ S∂L S ,
(2.6)
=−
d x ∂ H
∂L HM N −
4
d2
8
Customarily, one fixes static (Monge) gauge, f µ = 0, on the xµ (σ) and harmonic gauge on HM N .
–9–
where gauge dependent terms drop out. To obtain the contribution to the equation of
⊥
motion from the second line, express the variations δHN
M and δS in terms of δHM N using
the projection operators in Appendix A. The linearized Einstein-Hilbert operator becomes,
(Ed H)M N =
1
d−2
⊥
d HM
(∂M ∂N − ηM N d ) S .
N +
2
2d
(2.7)
Similarly, the induced brane field hµν is decomposed as
hµν = h⊥
µν + ∂µ aν + ∂ν aµ + ∂µ ∂ν φ +
1
ηµν s ,
4
(2.8)
⊥µ
= 0 and ∂ µ aµ = 0. h⊥
where ∂ µ h⊥
µν and s are gauge invariant. The brane
µν = 0, hµ
brane and operator (E h)
Einstein-Hilbert action SEH
4 µν can be read off from (2.6) and (2.7) for
d = 4, after appropriate field replacements. There are also new gauge invariant variables
based on f µ and y i to be introduced later in equations 2.17 and 5.7.
2.3 Thick branes and “blurred” quantities
Solving the equations of motion involves the massless scalar propagator in d dimensions,
G(x|| −
x′|| , x⊥
−
x′⊥ )
=−
Z
4
d k
Z
n
d q
ik(x|| −x′|| )+iq(x⊥ −x′⊥ )
e
k2 + q 2 − iǫ
.
(2.9)
kµ and qi denote momenta parallel and transverse to the brane. For two points restricted to
lie on a zero thickness brane, xi⊥ = x′i⊥ = y0i and the q integral in G(x|| − x′|| , 0) diverges for
n > 1. Correspondingly, a bulk field S sourced by a brane localized source δ(x⊥ −y0 )T (x|| ),
R
i.e., S(x|| , x⊥ − y0 ) = d4 x′|| G(x|| − x′|| , x⊥ − y0 )T (x|| ) also diverges as x⊥ → y0 . Hence,
the restriction of such bulk fields to the brane is ill defined. Essentially, a zero thickness
brane is a point source in transverse directions leading to a |x⊥ − y0 |−(n−2) divergence, as
in the Coulomb or Yukawa potentials.
This problem is resolved by realizing that dynamics give branes a form factor and
hence an effective thickness, as discussed in [43] for D-branes and in [17] for solitonic branes.
This is taken into account by replacing δ(x⊥ − y0 ) by a brane thickness profile function,
R
P (x⊥ −y0 ), with normalization dn x⊥ P = 1. Then, the divergent brane restriction S(x|| , 0)
of the bulk field is replaced by the “blurred” field hSi(x|| ), a weighted average over the
brane width,
Z
hSi(x|| ) =
dn x⊥ P (x⊥ − y0 )S(x|| , x⊥ − y0 ) ,
(2.10)
R
With Fourier transform conventions P (x⊥ − y0 ) = dn q Pe(q)eiq(x⊥ −y0 ) , one gets the momentum space equation
Z
n
e
e q) .
hSi(k) = (2π)
dn q Pe (q) S(k,
(2.11)
In the same way, the divergent brane restricted propagator G(x|| − x′|| , 0) gets replaced
by its well defined thick brane analogue,
Z
hGi(x|| − x′|| ) = dn x⊥ dn x′⊥ P (x⊥ − y0 ) G(x|| − x′|| , x⊥ − x′⊥ ) P (x′⊥ − y0 ) .
– 10 –
Physically, this is the propagation amplitude from x|| to x′|| located on the brane, with
a small delocalization in the transverse directions. This does not yet take into account
physical effects due to the presence of the brane. However, this construct will contribute
to the physical propagators and the interesting physics associated with it will be discussed
in the next section.
The corresponding momentum space expression (with Pe depending only on q 2 ) is
Z
[Pe(q)]2
e
hGi(k)
= −(2π)n dn q 2
.
(2.12)
k + q 2 − iǫ
Again, for a zero-thickness brane, Pe = 1 and the expression diverges for n > 1. For
thick branes, Pe (q) effectively implements a UV regularization at high q through a length
scale associated with the brane width, ω. Earlier, a non-zero brane width has been used
to justify implementing a sharp cutoff ∼ 1/ω on the q integral [23, 27], or solving the
equations separately outside and inside the thick brane [17, 34]. Here, we will explicitly
retain a smooth profile function Pe through which the results depend on the brane width
ω. Such dependences are not affected by the actual form of P , that encodes high energy
effects and which will only change the numerics.
Since the normalized P has mass dimension n, Pe (q) is dimensionless. Then rotational
invariance in the transverse space means it only depends on the combination q 2 ω 2 . This
implies (after simple manipulations in (2.12)),
e
hGi(k)
=
1
ω n−2
Σ −1 (ω 2 k2 ),
(2.13)
depending on k only through u = ω 2 k2 . The expression captures the form of divergence as
ω → 0. Later, Σ will appear in the same way as a self-energy correction in quantum field
e has the correct dimension for a propagator.
theory, hence the notation. Note that wn hGi
The use of the brane effective width ω means that the theory is valid for describing
interactions of transverse gravitons with the brane as long as q < 1/ω. Probing the brane
at shorter scales is meaningless in the low energy theory. To insure that interactions
too cannot probe beyond the brane width, the effective theory must also be restricted to
u = (kω)2 < 1, for momenta along the brane.
For the most part we do not need the functional form of P . But for explicit calculations,
a natural choice, motivated by [20], is the Gaussian form9 ,
P (x⊥ ) =
1
2
√
e−(x⊥ /2ω) ,
n
(ω 2π)
Pe(q) =
1
2 2
e−q ω /2 .
(2π)n
(2.14)
The profile function can also be discussed in a covariant setup, but that is not needed here.
2.4 Bulk-brane relations
For a thick brane, the pullback equation relating bulk and brane metric fluctuations becomes,
hµν = hHµν i + ∂µ fν + ∂ν fµ ,
(2.15)
In contrast with a sharp cut-off on q, a Gaussian Pe (q) leads to a non-negative P (x⊥ − y0 ), consistent
with its interpretation as a thickness profile. The main results are not affected by these choices.
9
– 11 –
with only x|| dependences. Note that fields intrinsically defined on the brane are not
affected by the “blurring” procedure (2.10). Then, using (2.5) and (2.8) gives,
1
1
⊥
⊥
hH iµν = hµν − ∂µ Fν − ∂ν Fµ − ηµν
hSi − s .
(2.16)
d
4
The F µ are new variables invariant under both bulk and brane gauge transformations,
1
Fµ = fµ + hAµ i − aµ + ∂µ (hΦi − φ) .
2
(2.17)
The gauge dependent variables in Fµ do not appear in the action (2.1). However, solving
the equations of motion requires either gauge fixing or using gauge invariant variables along
with (2.16). Thus, in bulk-brane setups, gauge variant variables survive through Fµ and
contribute to the solutions for the brane fields s and h⊥ . The implication of this as a broken
phase realization of 4-dimensional general covariance is discussed in the last section.
3. The Basic Brane Induced Gravity Model
In this section, we review and further investigate the solutions of the basic BIG model
[1, 2] based on bulk and brane Einstein-Hilbert actions of (2.1) but without SΛbrane and
brane . First, we solve the linearized equations of motion for the brane fields and review
SΩ
the tachyon/ghost problem [28]. The propagator is then analyzed in depth, extracting a
mass, a decay width and a modulation of the gravitational coupling. The suppression of
graviton decay into the bulk is explained with the help of the optical theorem. The reader
not interested in the details of solving the equations of motion can directly jump to the
solutions (3.15) and (3.16).
3.1 Equation of motion and solutions
For the basic BIG model, the linearized equation of motion for HM N expressed in terms
of Einstein-Hilbert operators (2.7) and for thick branes, takes the form
1 µν M N
MN
µν
δµ δν = 0 .
(3.1)
A(Ed H)
+ P (x⊥ − y0 ) B(E4 h)
+ T
2
In our momentum space conventions, it becomes10 (with pM = {kµ , q i }),
d−2 M N
MN 2
2
2
2 e ⊥M N
e
p p −η
(k + q ) S
−A (k + q )H
+
d
(k,q)
1
2
⊥µν
µ
ν
µν
2
µν
−Pe(q) B k e
h
+
k k − η k se − Te
δµM δνN = 0 .
2
(k)
(3.2)
This is to be combined with the surface equation (2.16). In the following, the iǫ terms
in the bulk propagator are not always written explicitly, but are finally included in the
blurred propagator (2.12).
10
The (µν) components alone can determine the brane fields, but it is more convenient to use all equations.
– 12 –
The ηM N -trace of (3.2) determines Se in terms of se,
e(q)
d
3
P
µ
2
e q) = −
S(k,
Teµ + B k se
.
A(d − 1)(d − 2) k2 + q 2 − iǫ
2
(k)
(3.3)
e in (2.12), one gets,
Restricting to the brane by using (2.11) and the expression for hGi
3
d
e
e
Teµµ + B k 2 se
hGi
.
(3.4)
hSi(k)
=
A(d − 1)(d − 2)
2
(k)
e q) in terms of hSi(k),
e
For later convenience, we express S(k,
e
1
e q) = − P (q)
e .
S(k,
hSi
2
2
e
k + q hGi
(3.5)
The (µ, i) components give (since “blurring” now involves integrating over an odd
function of q i ),
µ i
e q) ,
e ⊥µi (k, q) = − d − 2 k q S(k,
H
d k2 + q2
⇒
e ⊥ iµi (k) = 0 .
hH
(3.6)
However, note that using (3.5), and the transversality of H ⊥ , one gets,
e ⊥µi i(k) = −kν hH
e ⊥ iµν =
hqi H
gi µ e
d − 2 he
k hSi .
e
d hGi
(3.7)
where we introduce a new function he
g i(k) (that will drop out of most expressions),
n
he
g i = (2π)
Z
dn q
The (i, j) components give,
q 2 Pe 2
.
(k2 + q 2 )2
i j
e ⊥ij (k, q) = − d − 2 ( q q
e q) .
H
− η ij ) S(k,
d
k2 + q2
(3.8)
(3.9)
e ⊥ij i = 0 and, for i 6= j, hH
e ⊥ij i = 0. However,
From this one can immediately see that hqi H
!
he
g
i
gi e
d
−
2
⊥j
e ,
e ⊥ij i(k) = − d − 2 k2 he
e i(k) =
+ n hSi
hqi qj H
hSi . (3.10)
hH
j
e
e
d
d
hGi
hGi
The surface equation (2.16), on taking a trace and a double divergence, gives
4 e
1
1 e
⊥µ
µe
µ ν e⊥
2
µe
e
hHµ i = −2ik Fµ − hSi + se ,
k k hH iµν = −k 2ik Fµ + hSi − se . (3.11)
d
d
4
e µ⊥µ i = −hH
e ⊥i i and kµ kν hH
e ⊥ iµν = hqi qj H
e ⊥ij i(k), on using (3.10), (3.11) become,
Since hH
i
"
#
#
"
e
e
he
gi
se
hSi
hSi
he
gi
µ
µ
+ se , 2ik Fµ = (d − 2)
+ . (3.12)
+ n) − 4
−1
2ik Fµ = (d − 2)(
e
e
d
d
4
hGi
hGi
– 13 –
e and se as,
Eliminating kµ Fe µ gives a relation between hSi
(d − 1)(d − 5) e
3
hSi = − se .
d
4
(3.13)
Substituting back in the expression for kµ Feµ determines it in terms of se. Combining this
with the divergence of (2.16) and then using (3.7) leads to the solution for the Feµ fields,
!
µ
he
g
i
i
k
d
−
2
1
µ
n−3
Fe = −
U se , where , U =
.
(3.14)
e
4 k2
2 (d − 5)(d − 1)
hGi
Finally, combining (3.13) with (3.4) gives the solution (3.15) below for se in terms of T = Tµµ .
⊥ . One can then compute hH
⊥i
e µν
The (µ, ν) Components of (3.2) can be solved for Hµν
e e and F µ all of
using (3.5). Using (2.16), this becomes an equation for e
h⊥
µν in terms of S, s
which are known in terms of Teµν . Thus one has the final solutions,
se(k) = −
2
3B k2 +
1
1
e
h⊥
µν =
e −1
B k2 − A hGi
B
1
A d−2
B 2(d−5)
e −1
hGi
Te ,
1
kµ kν
Teµν − (ηµν − 2 ) Te
3
k
(3.15)
.
(3.16)
From this one can directly read off the gauge independent parts of the Greens functions
(0,2)
Gµνµ′ ν ′ , or the 4-dimensional gauge invariant amplitude,
Z
d4 x
Z
′ ′
d4 x′ T µν (x)Gµνµ′ ν ′ (x − x′ )T µ ν (x′ )
Z
Z
1
4
µν
4
µν ⊥
= d xT (x)hµν (x) = d x T hµν + T s .
4
(3.17)
If needed, the hµν in a given gauge can be constructed by adding gauge transformations,
1
hµν = h⊥
µν + 4 ηµν s + ∂µ ξν + ∂ν ξµ , and solving the gauge conditions for the ξµ .
One can immediately draw a number of conclusions, mostly known in the literature,
based on the structure of s and h⊥ solutions:
e n=0 =
(1) Zero thickness limit: Note that in the absence of extra dimensions, hGi
−1/k2 , leading to the solutions for ordinary Einstein-Hilbert gravity,
µ kν
1
1
k
2 1
1
⊥
(e
s)n=0 = − ′ 2 Te .
(3.18)
(e
hµν )(n=0) = ′ 2 Teµν − (ηµν − 2 )Te ,
A k
3
k
3A k
Also he
g in=0 = 0 gives (Feµ )n=0 = 0. The BIG results (3.15),(3.16) differ from this only in
e −1 dependent terms. Now, while the brane width ω was introduced to keep hGi
e
the hGi
e diverges (from
finite for n > 1, the final solution is well defined for ω → 0. Then, hGi
(2.13)), resulting again in (3.18). Hence, for n > 2, modifications of 4-dimensional gravity
arise only for thick branes. Since the spin 2 and spin 0 parts of the propagator in BIG
have the same coefficients as in ordinary gravity (except d = 5 where se = 0), there is no
vDVZ discontinuity [31, 32, 33] in the ω → 0 limit.
– 14 –
e 2 = 0) is finite and contributes a hard
(2) Tachyon problem: For n > 2, hGi(k
mass to the propagators. The scale of the mass is set by Aω n−2 /B and the gravitational
coupling is given by GN ∼ 1/B. Thus, for phenomenological reasons, B should be large
e −1 appears with opposite signs in the denominators,
and Aω n−2 should be small. But hGi
indicating that if h⊥ has a healthy mass, then s will be tachyonic or vice versa [28]. For
the conventional choice of B > 0 and A > 0, h⊥ cannot have a tachyonic pole (since for
e
any k2 > 0, (2.12) gives hGi(k)
< 0). Then, s is tachyonic.
3) Ghost problem: It is well known [31, 44, 45] that a propagator for hµν with the
tensor structure of ordinary massless gravity in 4-dimensions, is ghost free only for zero
graviton masses. Thus the non-zero masses and the −2/3 factor in the se solution makes
the tachyonic spin 0 mode also a ghost [28]. The presence of this tachyonic ghost in brane
induced gravity has so far hampered its further development. One of our purposes in this
paper is to shed light on the origin of this ghost mode.
4) Unstable gravitons: Ignoring the tachyonic ghost se field, the massive spin 2
graviton h⊥ on the brane is not a stable particle but a resonance state. Technically, this
e has a branch cut from k2 = 0 to −∞ due to the continuum of Kaluzais because hGi
Klein modes resulting from an uncompactified bulk ([5, 6]). For very small masses, the life
time is very large. In the next subsection we will explain the origin of the decay and its
suppression using standard quantum field theory and then carry out a detailed analysis of
the propagator.
5) String theory limit: The string theory computation of the BIG action in [18, 19]
obtained B < 0, making h⊥ the tachyonic ghost. But in this setup, in terms of the string
coupling gs , A ∼ 1/gs2 and B ∼ 1/gs . Assuming ω ∼ gs0 , weak string coupling corresponds
to large masses, beyond the validity limits of the theory. Even for D3-branes where n = 6
and ω 4 ∼ gs is a more natural choice, masses are gs -independent and at string scale. In
either case, for k2 values within the validity range of the theory, the propagator is given
e
by −hGi(k)/A,
avoiding the tachyon/ghost problems, and gravity is essentially higher
dimensional. However, the orbifold based construction of [20] that attempts to derive BIG
from string theory with the phenomenologically interesting parameter ranges, does not
evade the tachyon/ghost problem in this manner.
In the rest of this section we concentrate on the healthy spin 2 mode, returning to the
origins of the ghost mode in the next section.
3.2 Graviton decay from optical theorem
A main feature of BIG, analyzed around flat background, is that h⊥ is a massive unstable
graviton in 4 dimensions due to the possibility of gravitons escaping into the bulk [1, 5, 6].
However, the graviton lifetime must be large for phenomenological reasons. Below, we
show that the suppression of graviton decay can be explained using general field theory
arguments. The detailed structure of the propagator is analyzed in the next subsection.
The bulk propagator G(x − x′ ) (2.9) gives the amplitude for graviton propagation
from any x to any x′ (ignoring the tensor structure and for canonically normalized fields,
so no 1/A factor). But hGi(x|| − x′|| ) is constructed to describe a restricted propagation
between two points with their x⊥ coordinates delocalized over the width ω of a thin region
– 15 –
that would be occupied by the brane. Of course, there is a finite probability σescape that
gravitons emitted at x|| within the thin region, end up somewhere in the bulk. Such
processes are not described by the restricted propagator hGi and appear to it as decay
channels, thus giving it an imaginary part. The relationship is quantified by the optical
e −1 (with
theorem where, in terms of the corresponding amputated 2-point function ω −n hGi
ω factors restoring the correct dimensions for the propagator),
e −1 ) ∼ σescape 6= 0 .
2 Im (ω −n hGi
This is a geometric result that holds for any region of space that would be occupied by a
thick brane and applies to gravitons not yet dynamically affected by a physical brane.
Now, with a physical brane in place and after taking gravitational dynamics on the
e −1 )−1
ebb = −(k2 − A hGi
brane into account, we obtained a brane-to-brane propagator G
B
in (3.16), but now written for the canonically normalized brane field and with suppressed
tensor structure. The stability of the corresponding state is again encoded in the imaginary
part of the amputated 2-point function which now gives,
n
e −1 ∼ Aω σescape .
e−1 = 2 A Im hGi
(3.19)
2Im G
bb
B
B
Hence, graviton decay on the brane is still due to escape into the bulk, but with an amplitude suppressed by a factor Aω n /B as compared to the unhindered escape in the absence
of brane gravitational dynamics. This is a way of seeing that a large brane Einstein-Hilbert
term confines gravitons to the brane by suppressing their escape into the bulk.
Finally, for completeness, one can also compute the brane-to-bulk propagator,
"
#
e(q) hGi
e −1 1
1
−
P
eBb (k, q) =
.
G
e −1
k2 + q2
B k2 − A hGi
B
The brane-to-bulk decay probability σbrane→bulk can be constructed out of this after proper
amputations (using Gbb for the brane leg and G for the bulk leg). One can then verify the
optical theorem for the physical propagator,
e−1 ∼ σbrane→bulk
2Im G
bb
In this sense, the verification of the optical theorem on the brane requires using the braneto-bulk propagator, since the brane theory alone is not unitary.
3.3 Graviton mass, decay width and coupling
Let us investigate the properties of the spin 2 graviton h⊥ (3.16) in more detail. The physics
e 2 − iǫ).
of the corresponding brane-to-brane propagator follows from the structure of hGi(k
We will see that the propagator has the standard form for a massive unstable particle in
QFT. More importantly, for small masses it is amenable to the standard particle physics
approximation methods.
e (2.12) we use the Gaussian form (2.14) for Pe(q). Doing the angular
To evaluate hGi
integrals in q-space gives,
Z
−q 2 ω 2
n
n
e = −2N dq q n−1 e
,
N −1 = (4π) 2 Γ( ) .
(3.20)
hGi
2
2
k + q − iǫ
2
– 16 –
First, let us consider the even n case and set 2m = n − 2 (the result for odd n is given
later). After some manipulations,
m (−1)m N m
∂
u
e
hGi(k) = −
e
E
(u
−
iǫ)
,
(3.21)
u
1
ω n−2
∂u
where u = ω 2 k2 and we have used the notation [46],
Z ∞
e−t
.
dt
E1 (u − iǫ) =
t
u−iǫ
(3.22)
e one gets
Since ∂u E1 (u) = −eu /u, for Σ−1 = ω n−2 hGi
#
"
m
X
(−1)r (r − 1)! um−r .
Σ−1 (u − iǫ) = −(−1)m N um eu E1 (u − iǫ) +
(3.23)
r=1
The corresponding expression for odd n is given by equation (3.35) below.
Remember that the iǫ prescription tells us how to handle the k2 + q 2 = 0 poles in
(2.12). Now, after performing the q-integrations, the same prescription will dictate the
correct ǫ → 0 limit. To see what happens, note that the function E1 (z) is the analytic
continuation, to the complex plane, of the exponential integral Ei(−u) defined, for real u,
by
Z ∞
e−t
dt
,
(3.24)
Ei(−u) = −
t
u
Ei(−u) is defined over the entire real line, but E1 (z) has a branch cut from z = 0 to
z = −∞ along the negative real axis. The crucial point is that as ǫ → 0, we approach the
branch cut from below and
lim E1 (u − iǫ) = −Ei (−u) + iπ θ(−u) ,
(3.25)
ǫ→0
where, θ(−u) is the unit step function. The imaginary part is the origin of the complex
pole in the propagator, and hence of graviton decay 11 . Writing Σ−1 = a + ib, the real and
imaginary parts are given by
#
"
m
X
r
m−r
m
m u
,
(3.26)
(−1) (r − 1)! u
a(u) = (−1) N u e Ei(−u) −
r=1
m −|u|
b(u) = −π N |u| e
θ(−u) .
(3.27)
Finally, in terms of Σ = Σ1 + iΣ2 , the brane-to-brane propagator in the h⊥ solution (3.16)
is (with canonical field normalization and suppressed tensor structure),
ebb =
G
11
k2 −
A ω n−2
B
h
−1
Σ1 (ω 2 k2 ) + iΣ2 (ω 2 k2 )
i,
(3.28)
Approaching the branch cut from above (corresponding to E1 (u + iǫ)), would give −iπ which does not
lead to decay. So the correct sign is dictated by the iǫ prescription in the bulk propagator.
– 17 –
The above analysis then gives,
Σ1 (u) =
a
,
a2 + b2
Σ2 (u) =
−b
≥ 0.
a2 + b2
(3.29)
This propagator has the familiar quantum field theory form where the analogue of Σ
arises due to self-energy corrections. It has a complex mass pole and, strictly speaking, the
graviton mass and decay width are given by the real and imaginary parts of the complex
pole. In practice, determining the location of the complex pole is not easy. But since the
determination of mass anyway becomes somewhat arbitrary due to the finite life-time, an
approximation can be made provided, near the mass pole, |Σ1 | is much larger than |Σ2 |.
In such cases, a mass can be defined as the pole position with only Σ1 present, while Σ2
gives the decay width (see, for example, [47, 48]). This is often a valid approximation in
perturbative quantum field theory. Fortunately, one can show that it also holds here for
small enough masses: For m ≥ 1, as u → 0, Σ2 vanishes as um while Σ1 goes to a constant
−1/[N (m − 1)!]. Hence, for a small enough mass, at the mass pole M 2 = −u/ω 2 , one has
|Σ1 | >> |Σ2 |, as wished.
Then, in this approximation the mass M 2 = −k2 is determined, on ignoring Σ2 , by
B
ω 2 k2 = Σ1 (ω 2 k2 ) .
(3.30)
Aω n
For n > 2, Σ1 (0) = −[N (m − 1)!]−1 < 0. So, by continuity, Σ1 (ω 2 k2 ) < 0 for small
arguments in the neighborhood of zero. Then the graviton is non-tachyonic, M 2 > 0.
More concretely, this is visible in Fig.1(a) where the right hand side of the above equation
is plotted against u = ω 2 k2 for n = 6, 5, · · · , 3 (odd n curves are based on (3.35)) and the left
hand side is plotted for two different slopes B/Aω n , with values chosen only for illustrative
purposes. The intersections determine the masses M 2 = −u/ω 2 , with larger slopes resulting
in smaller masses. The plots also show that Σ1 is monotonic for n > 2 and the propagator
has a single mass pole. In Fig.1(b) we plot Σ2 vs ω 2 k2 . Comparing curves for the same
n in both plots again shows that for very small masses, |Σ2 (−ω 2 M 2 )| << |Σ1 (−ω 2 M 2 )|,
justifying the approximation (the curves for n = 1, 2 look rather different and are not
shown here).
Now, in terms of the mass as determined above, the propagator has the standard form,
where,
ebb = −
G
Z(k)
n−2
k2 + M 2 − i AωB
Z −1 (k) = 1 −
Z(k) Σ2 (k)
,
(Aω n−2 /B) Σ1 (ω 2 k2 ) + M 2
.
k2 + M 2
(3.31)
(3.32)
This will look more familiar when Σ1 (u) is Taylor expanded around u = −ω 2 M 2 .
Note that Z retains a momentum dependence which amounts to a momentum dependence of the effective gravitational coupling,
f
Gef
N (k) = Z(k)/B .
– 18 –
(3.33)
(a) Determination of Mass
(b) Decay Width
Figure 1: Behaviour of Σ1 (u) (left) and Σ2 (u) (right) for codimensions n = 6 (the lowest curves)
through n = 3 (close to the x-axis)
This momentum dependence can be expressed entirely in terms of the dimensionless parameters u and Aω n /B. In particular, for time independent sources (k2 > 0), Σ2 = 0 and
this is the only modification besides the graviton mass.
A simple analysis of Z(k) shows that at large momenta, |k2 | >> M 2 , Z ∼ 1. But for
small momenta, |k2 | ∼ M 2 , on Taylor expanding around the mass pole given by (3.30), one
gets Z < 1. Hence the gravitational coupling is weaker at large distances, but the variation
is not drastic. This is over and above the effects due to graviton mass and decay width.
Now, comparing (3.31) to the relativistic Breit-Wigner form near the mass pole, the
decay width is identified as
Σ
(u)
Aω n−2
2
Z(u) Σ2 (u)
=
−M
Z(u)
Γ=
MB
Σ1 (u) u=−ω 2 M 2
> 0.
(3.34)
u=−ω 2 M 2
Once again, note that it is the iǫ prescription in the bulk propagator that leads to the
correct sign for the decay width.
For odd n the analysis proceeds along similar lines, except that the exponential integral
is replaced by the error function and, with 2m′ = n − 1, we get,
h ′ 1
√
′
Σ−1 (u − iǫ) = −(−1)m N π um − 2 eu 1 − erf ( u − iǫ)
′
m
(2r − 3)!! m′ −r i
1 X
(−1)r
u
.
+√
π r=1
2r−1
– 19 –
(3.35)
In fact, this expression has been used to
plot the curves for n = 3, 5 in Fig.1(a)
and Fig.1(b). A unified description of both
even and odd n is given in terms of the
Whittaker functions,
n
Σ
−1
1
u 4 −1 e 2 u
W− n , n−2 (u − iǫ) .
=−
n
4
4
(4π) 2
For completeness, in Fig.2, we show a
representative plot for the spectral density
function,
1
ρ(s) = − ImGbb (k2 = −s) ,
π
to show that it too has the standard form
for an unstable particle, again with the
Figure 2: A Sample Spectral Density Function
Aω n /B value chosen only for the purpose
of illustration.
To summarize, we have shown that although the origin of the “self-energy” contribution, Σ, to the propagator in the BIG setup is very different from its origin in perturbative
quantum field theory, the resulting unstable massive particles behave in very similar ways
and are amenable to the same approximation. Hence, the modification of the spin 2 gravie −1 , breaks up into 3 contributions: a hard mass M
ton propagator (3.16), contained in hGi
given by (3.30), a decay width Γ (3.34) and a momentum dependent Newton “constant”
(3.33).
4. Screening of Λ and the Origin of Tachyon/Ghost Problem
The possible resolution of the cosmological constant problem is the main reason for interest
in n > 2 brane induced gravity, despite its ghost/tachyon issues. Support for this expectation comes, as discussed in [5, 6], from the classical solutions of d-dimensional gravity
[14, 15] that describe a 3-brane with an ADM mass density corresponding to a non-zero
brane tension (which is the same as the brane cosmological constant). To be precise, the
classical solutions describe this setup at distances far away from the location of the brane
core. The interpretation is that, for codimensions n > 2, a brane cosmological constant Λ
can curve directions transverse to the brane rather than those parallel to it, keeping the
brane metric flat12 . Thus the effect of Λ is absorbed by bulk curvature and it does not
curve the brane worldvolume to a de Sitter spacetime.
Now, the presence of a brane Einstein-Hilbert term in the action does not affect this
classical solution. But for a large enough coefficient, it is expected to dictate the dynamics
of metric perturbations hµν around the classical solution. This is why one expects n > 2
12
[5, 6] also consider time dependent solutions but these do not show up in our linearized analysis and
are not discussed here.
– 20 –
BIG models to describe effectively 4-dimensional gravity with suppressed Λ (The n = 2
model is somewhat special and has been analyzed in some detail in [24, 25, 26]).
Unfortunately, implementing the above scenario is not as straightforward. In reality, to
study metric perturbations hµν , one needs classical solutions that are valid in the vicinity
of the brane (at least with some effective thickness included). The existing solution [14, 15]
is not adequate for this purpose since the energy momentum tensor that sources it near
the brane core, does not correspond to a brane Λ source.
Hence, in practice, in BIG with codimension n > 2, the analysis of metric perturbations
has mostly been performed in a flat background, instead of one sourced by Λ (even though
the latter is used to argue for the suppression of Λ). It is also in the flat background
analysis that one encounters the tachyonic ghost problem reviewed in the previous section.
One may conjecture that this problem is an artifact of the flat approximation and would
go away once a full fledged analysis is performed around a curved background sourced by
Λ. However, in the absence of a classical solution adequate for such analysis, here we point
out that the flat background analysis of the previous section is, by itself, sufficient to study
the response of gravity to a small brane cosmological constant and exhibit its suppression.
This also has ramifications for what to expect from a curved background analysis for the
resolution of the ghost issue (as will be detailed below). The healthy spin 2 part of the
graviton, on which we have concentrated so far, is not relevant here. Below we return to
the unhealthy scalar mode to find that,
• The “tachyonic” mass is actually healthy and is instrumental in filtering out Λ in
the flat background approximation. It arises, at least partly, as an artifact of the flat
approximation and as a substitute for the background curvature.
• For phenomenologically interesting values of parameters (corresponding to a small
graviton mass), the ghost nature is more intractable and will most likely persist even
when metric perturbations are analyzed in the background curved by Λ, unless one
ends up with massless gravity on the brane in a fully non-linear background.
• The 4-dimensional Newton constant and other couplings are modified by the screened
Λ, consistent with [3, 5, 6]. This may help in avoiding a very low bulk gravity scale
A, but may introduce hierarchy problems for other couplings.
• Finally we emphasize that BIG and massive Fierz-Pauli gravity (as studied in [38])
use different mechanisms to respond to Λ, in spite of a superficial similarity.
4.1 Filtering out of brane cosmological constant
It is easy to see that brane induced gravity for n > 2 is not very sensitive to a brane
cosmological constant: The addition of the brane tension/cosmological constant term,
p
R
SΛ = −Λ d4 σ −g(X(σ)), to the action amounts, at the linearized level, to the shift
Teµν (k) → Te ′µν (k) = Teµν (k) + Λ ηµν δ(4) (k) .
(4.1)
The validity of the perturbative analysis around flat background requires that Λ is small, of
the same order as Tµν . In ordinary Einstein-Hilbert gravity, Λ, however small, destabilizes
– 21 –
the flat space solution toward a de Sitter space (the manifestation of this at the propagator
level is described below). Hence, in a modified theory of gravity the stability of flat space
in the presence of Λ is an indication that gravity has been made less sensitive to the
cosmological constant. The analogue of this argument for Fierz-Pauli massive gravity is
considered in [38], although the actual mechanisms differ.
Let us write the solutions (3.15) and (3.16) explicitly in terms scalar and tracelesstransverse components of Tµν ,
se(k) = −
2
3B k2 +
1
A d−2
B 2(d−5)
e −1
hGi
Tes ,
1
1
e
Te⊥ .
h⊥
µν =
A
e −1 µν
B k2 − hGi
B
(4.2)
As discussed in Appendix A, it is only in this form that the solutions are valid for a
⊥ = 0 and T
eΛs = 4Λ δ(4) (k), leading
cosmological constant source. For such a source, TeΛµν
to e
h⊥ = 0 and,
Λµν
seΛ (k) = 4 c δ(4) (k) ,
where
c=−
4 d−5 Λ e
hGi(0) .
3 d−2 A
(4.3)
e
The crucial point is that only for n > 2, hGi(0)
is finite and the analysis well defined
n
e
(for example, for a Gaussian profile and for even n, hGi(0)
= − ( n2 − 2)!/[(4π) 2 Γ( n2 )ω n−2 ]).
Thus, in coordinate space, Λ effectively shifts the background from s = 0 to a constant value
s = 4c, keeping the flat background essentially unchanged. It is instructive to contrast this
with Einstein-Hilbert gravity where one would get seΛ ∼ Λ δ(4) (k)/k2 , which is a solution
to 4 sΛ ∼ Λ. Then sΛ is quadratic in distance and diverges at large distances, indicating
that flat space is not a good starting point for a perturbative expansion of de Sitter space.
In contrast, in BIG, Λ does not produce a non-flat metric on the brane and is hence filtered
out 13 .
The mechanism by which BIG filters out Λ is sometimes heuristically explained on the
basis of the behaviour of the n = 1 model in which gravity becomes higher dimensional,
and hence weaker, at very large distances. But this model does not filter out Λ and the
n > 2 model that does, has a different large distance behaviour. So let us reiterate two
relevant aspects of the above analysis:
e
1. Only a non-zero hGi(0)
is relevant to filtering out Λ. The momentum dependence of
e
hGi (that contributes to graviton decay and a varying gravitational coupling) does
not play any role in this, in contrast to the more general “filter” mechanisms of [37]
2. It also follows that at large distances (k2 → 0), the behaviour of gravity does not
become higher dimensional, unlike the n = 1 DGP model. Hence the filtering of the
cosmological constant cannot be attributed to such a behaviour.
In BIG, the cosmological constant is completely filtered out at the linearized level. One
needs to go beyond the linear analysis to make a statement about its remnant effects. In
particular, explaining the small observed value of Λ (or dark energy density) is a different
issue not addressed by BIG at this stage.
e
Codimensions n = 1, 2 give hGi(0)
= ∞ due to an IR divergent q-integral and the above discussion of
filtering, formulated around flat background, does not apply. For the the n = 2 case, see [24, 25, 26].
13
– 22 –
4.2 On the origin of the tachyon and ghost problems
As we have seen, a tachyonic ghost mode is encountered while analyzing BIG around flat
background [28]. Although flat space is a valid solution to the Λ = 0 model, one may still
suspect that the tachyon and ghost problems are related to this very set up. After all, a
generic 3-brane always has a tension which, in some sense, is the stuff that holds the brane
together. So one always has to work with the full action,
bulk
brane
brane
SEH
+ SEH
+ SΛbrane + Sm
.
R
√
Even though a Λ → 0 limit may exist, in that limit SΛbrane = Λ d4 σ −g should be
replaced by the action for a tensionless brane [49, 50] rather than totally eliminated. But
tensionless branes have special properties that make them unsuitable for our purposes and
hence need not be considered. This line of argument tells us that the “basic” brane induced
gravity model reviewed in section 3 does not really correspond to a brane setup, unless
augmented by a brane tension term. Could this also provide a solution to the tachyon and
ghost problems? This possibility is discussed below.
One approach to addressing this question is to consider, as suggested in [3, 5, 6], a
brane can be neglected to first approximation. This
large SΛbrane so the matter action Sm
leads to classical solutions for bulk gravity sourced by Λ. Around this one should study
brane and influenced by S brane . As pointed out above,
metric perturbations sourced by Sm
EH
the existing solutions [14, 15] are not adequate for this purpose since they do not satisfy
the equations of motion with the correct brane source. Hence, such analysis has not yet
been carried out (see footnote 3).
brane ) is not the only one to explore.
However, the above parameter range (SΛbrane >> Sm
In fact, we saw that the linearized analysis is already capable of exploring the model in the
brane have comparable but small contributions. Thus, it can also
regime where SΛbrane and Sm
provide some insight into the implication of the brane tension for the tachyon and ghost
problems.
The tachyon and ghost problems showed up in the propagator (3.15) for the s field
which has the general structure
−
1
2
.
2
n−2
3 Bk − A ω
|N (0) | Σ(k)
The overall sign signals a ghost and the relative sign in the denominator, a tachyon. The appearance of A and B coefficients (associated with bulk and brane actions) shows that Σ(k)
(relevant to the tachyonic nature) arises entirely from integrating out bulk modes, while the
rest (the −2/3 factor and hence the ghost nature) originates in the brane Einstein-Hilbert
term. Let’s emphasize that the above form is a valid way of presenting and interpreting the
propagator as long as the B term in the action dominates over the A term. This is the phenomenologically interesting case, with a very light graviton and effectively 4-dimensional
gravity. It is also in this range that the tachyon and ghost identifications hold 14 .
14
For example, in the opposite case when the A term dominates over the B term (as in string theory
setups), the propagation is described by the Σ(k) term and gravity is healthy and essentially d-dimensional.
– 23 –
• The tachyon problem: Now, this is not hard to dismiss. In the previous subsection
we saw that the “tachyonic” mass of s (more precisely, the Σ(0) 6= 0 part) was
instrumental in “absorbing” Λ and keeping the brane metric flat. On the other
hand, the classical solutions of [14, 15] indicate that Λ can equally well be absorbed
by curving directions transverse to the brane, keeping the brane metric flat. The
comparison indicates that approximating the bulk as flat, has to force a mass on s in
order for the brane metric to remain flat in the presence of Λ. In this sense, the mass
is at least partly an artifact of the flat background approximation and is functionally
equivalent to a “healthy” bulk curvature to be sourced by Λ. This demonstrates that
the “tachyonic” mass is not an unhealthy feature. Indeed, m2s is tachyonic only with
respect to the ghost-like kinetic term of s, and as such has a positive contribution to
R
the s field Hamiltonian ∼ (−ṡ2 − ∇s2 + m2s s2 ).
• The ghost problem: This however is more fundamental and may not be cured even
if the flat background is replaced by a curved one sourced by the brane tension . The
reason is that even in such a curved background, the brane metric remains flat and
brane , expanded around flat 4-dimensional
hence its fluctuations are described by SEH
spacetime. This contributes a factor −(2/3)(1/Bk 2 ) to the scalar mode propagator,
exactly as in flat background. A modified “self-energy” term, Σcurved , arises from
integrating out bulk modes, but now it also depends on the brane tension. As before, small graviton masses and effectively 4-dimensional gravity require a large B
coefficient. Thus the overall propagator again has the standard 4-dimensional tensor
structure, corrected by self-energy terms. Now, it is well known that a propagator
with a tensor structure associated with the 4-dimensional Einstein-Hilbert action is
ghost free only in the massless limit (where the scalar mode ghost cancels against a
contribution from the spin 2 mode). As soon as the mass poles are shifted away from
zero, the scalar mode ghost remains uncanceled.
An implication is that the ghost can be avoided only if Σcurved (0) = 0. In a flat
space background, this would have prevented the theory from filtering out the cosmological
constant. But in a curved background, where Λ is already absorbed in the background
curvature, this could lead to a consistent theory of massless gravity with filtered out Λ.
However, it is not obvious that a background sourced by the brane tension will lead to
such a self-energy contribution. In fact the opposite seems to be the case by a continuity
argument: In the small Λ limit, the expressions should reduce to what we have already
computed, in which case Σ(0) 6= 0. Hence we conclude that in the phenomenologically
interesting parameter ranges, the scalar ghost will be a generic feature of brane induced
gravity in any background, except those for which Σcurved (0) = 0.
The absence of a ghost in the d = 5 DGP model is a numeric coincidence. On general
grounds, reduction of gravity from d to 4 dimensions gives a factor d−5 in the scalar sector.
In the BIG setup, this results in solutions with s = 0, thus evading the ghost problem.
4.3 Implication for couplings
The structure of the solution (4.3) essentially corresponds to a warped metric with its
– 24 –
standard implications for the couplings. The metric in the presence of Λ and matter
sources becomes,
(m)
gµν (x) = (1 + c) ηµν + hµν
(x) ,
(4.4)
(m)
where c ∼ Λ/(A ω n−2 ) > 0 and hµν is the metric perturbation sourced by matter. Adopting the standard convention to use ηM N as the flat spacetime metric, one defines the
physical metric
gµν
′
.
(4.5)
gµν
=
(1 + c)
R√
R√ ′ ′
R√
In terms of this, B
gR = (1 + c)B
g R . For the matter action,
gL(g, ψ, λ) =
R
√ ′
2
′
′
′
g L(g , ψ , λ ), where the matter fields, collectively denoted by ψ, and their
(1 + c)
couplings λ, have to be scaled appropriately for the equivalence principle to hold. This
results in an effective B ′ = B/(1 + c) or an effective Newton constant G′N = (1 + c)GN ,
consistent with related observations in [17, 5, 6, 3, 25, 26]. Although this analysis is valid
for small c (so that our perturbative treatment remains valid) it indicates that for a large
Λ one needs a starting B larger than the observed Mp2 (for c > 0). This can relax the
constraint on the bulk coupling A based on the smallness of the graviton mass by a factor
(1 + c) which is welcome news. However, it may also introduce new hierarchy issues for
other standard model couplings.
4.4 Contrast with Fierz-Pauli Massive gravity
In the ghost-free FP massive gravity theory around flat background, reviewed in a later
section, the spin-2 graviton h⊥ has a mass m2 , whereas, to avoid the ghost, the mass m0 of
the scalar mode s is sent to infinity. Hence for any matter source Tµν , s = 0. The caveat is
that a constraint equation one gets for s (for m0 = ∞) only implies 4 s = 0 so, in general,
s = c1 + c2 sharmonic . Substituting back into the equations, gives c2 = 0 for any source
and c1 = −4ΛGN /3m22 6= 0 only for a cosmological constant source. To contrast the two
theories, in BIG, s is always sourced and remains well behaved in the presence of Λ due
to its own mass, but in the ghost free FP gravity, s is zero except for a Λ source and the
screening parameter is the mass of spin-2 graviton m2 , which is itself not sourced by Λ! A
FP theory with finite m0 (and hence a ghost), when sourced by Λ, would behave similar
to BIG for n > 2, with m0 as the screening parameter. However, the m0 → ∞ limit is not
continuous. Thus, although superficially similar, BIG with n > 2 and FP massive gravity
use different mechanisms to filter out Λ.
5. Inclusion of Extrinsic Curvature Terms
We now solve the modified equations of motion after the inclusion of the extrinsic curvature
terms. These are ubiquitous in any brane induced gravity setup and arise at the same
perturbative order as the induced Einstein-Hilbert term [18, 19, 52]. The brane transverse
fluctuations enter as new degrees of freedom. The brane tension can be included in Tµν .
Again the analysis is performed in terms of gauge invariant variables. It turns out that the
extrinsic curvature terms have no effect whatsoever on the solutions for the brane fields.
While this does not cure the tachyon/ghost problem, it at least insures that 4-dimensional
– 25 –
gravity is not modified in other unwanted ways by the extrinsic curvature terms. The
negative result is essentially due to the Z2 reflection symmetry of the background about
the brane position. Breaking this symmetry will lead to extra non-trivial contributions
from the extrinsic curvature terms. With this summary, the reader not interested in the
details of the calculation can safely skip this section.
5.1 Extrinsic curvature term in the action
The contribution of the extrinsic curvature to the action is [18]
Z
√
αβ
β
Mα
,
Ω
−
Ω
Ω
SΩ = C d4 σ −g ΩM
α
αβ
M
Mβ
(5.1)
where
M
N
K
M
λ
+ ΓM
ΩM
N K ∂α x ∂β x .
αβ = ∂α ∂β x − γαβ ∂λ x
(5.2)
Here γ and Γ are the Christoffel connections corresponding to the metrics gαβ (x(σ)) and
GM N (x(σ)), respectively. For a thin brane the quadratic action for the fluctuations can be
worked out by setting g = η + h, G = η + H and δX M = {f µ , y i },
Z
h1 SΩ = C dx4||
∂α Hiβ ∂ α H iβ − ∂α H αi ∂ β Hβi + ∂α H αi ∂i Hββ − ∂α H iβ ∂i H αβ
2
i
1
+ ∂i Hαβ ∂ i H αβ − ∂i Hαα ∂ i Hββ + y i ∂i 4 Hαα − ∂α ∂β H αβ + · · · , (5.3)
4
where the ellipsis stand for functions of hµν − Hµν that do not contribute to the equation
of motion for a metric variation δhµν = δHµν .
As described in section 2, for thick branes, HM N (x(σ)) are replaced by hHM N i(x|| ).
A new feature of SΩ is that it contains transverse derivatives of H restricted to the brane,
∂i H|brane ≡ ( ∂x∂i H)(x|| , xi⊥ = y0i ). For a thick brane, these should be replaced by h∂i Hi ≡
⊥
h ∂x∂i Hi(x|| ). The y i (x|| ) are brane fields and do not need any “blurring”. Hence, for thick
⊥
branes, the appropriate action is,
Z
h1 ∂α hHiiβ ∂ α hHiiβ − ∂α hHiαi ∂ β hHiβi + ∂α hHiαi h∂i Hiββ
SΩ = C dx4
2
1
−∂α hHiiβ h∂i Hiαβ +
h∂i Hiαβ h∂ i Hiαβ − h∂i Hiαα h∂ i Hiββ
4
i
+y i 4 h∂i Hiαα − ∂α ∂β h∂i Hiαβ + · · · .
(5.4)
5.2 Equations of motion with extrinsic curvature contributions
To obtain the contribution of SΩ to the equations of motion for a thick brane, one considers
variations δy i , δhHi, δh∂i Hi and re-expresses the last two in terms of the bulk variation
δH. In particular,
Z
δh∂i HiM N =
dn x⊥ P (x⊥ − y0 ) ∂i δHM N (x|| , x⊥ − y0 )
Z
= − dn x⊥ ∂i P (x⊥ − y0 ) δHM N (x|| , x⊥ − y0 ) .
(5.5)
– 26 –
Then for δH the variation can be written in a compact from as,
Z
1
iαβ
d
iαβ
δHαβ ,
δSΩ = −2C d x P (x⊥ − y0 ) ∂α N
δHiβ − ∂i P (x⊥ − y0 ) N
2
(5.6)
where, to linear order,
1
i
Nαβ
(x|| ) = hΩiαβ − Ωiλ λ ηαβ i = ∂α ∂β y i +
∂α hHiiβ + ∂β hHiiα − h∂ i Hiαβ
2
1 i λ
i
λi
− 4 y + ∂λ hHi − h∂ Hiλ ηαβ
2
1
∂α hH ⊥ iiβ + ∂β hH ⊥ iiα − h∂ i H ⊥ iαβ
= ∂α ∂β F i +
2
1 i ⊥ λ
3
i
⊥ λi
− 4 F + ∂λ hH i − h∂ H iλ ηαβ + ηαβ h∂ i Si .
2
2d
F i are gauge invariant extensions of y i (the analogues of F µ in transverse directions),
1
F i = y i + hAi i + h∂ i Φi .
2
(5.7)
bulk + S brane + S brane ) one obtains the thick brane metric equation of motion
Now, from δ(SEH
EH
Ω
with extrinsic curvature contributions contained in the C-dependent terms,
A
d−2 M N
MN
⊥M N
∂ ∂ −η
d S
d H
+
2
d
"
#
1
B
i
4 h⊥µν + (∂ µ ∂ ν − η µν 4 ) s δµM δνN − C ∂ λ Nλν
+P (x⊥ − y0 )
(δiM δνN + δνM δiN )
2
2
1
+ C ∂i P (x⊥ − y0 ) N iµν δµM δνN = − P (x⊥ − y0 ) T µν δµM δνN .
2
(5.8)
Also, in the quadratic action SΩ , the y i appear as Lagrange multipliers and their
equation of motion is the constraint equation,
1 i
3
µ ν i
⊥λ
µ ν ⊥
∂ ∂ Nµν ≡ h∂ 4 Hλ − ∂ ∂ Hµν + 4 S i = 0 .
(5.9)
2
d
But we will see that this is already contained in (5.8). The expression within braces is the
scalar curvature of the metric (η + H)µν .
5.3 Solutions with extrinsic curvature contributions
Below we show that the inclusion of C-dependent terms in (5.8) has no effect on the
solutions for brane fields. We closely follow the steps in the C = 0 case.
i . On transforming to momentum
The ηM N -trace of (5.8) gives S in terms of s and Nµν
space, one gets the solution
e q) =
S(k,
Pe(q)
−d
A(d − 1)(d − 2) k2 + q 2
3
µ
2
iλ
e
e
Tµ + B k se + i2C qi Nλ
.
2
– 27 –
(5.10)
e i λ depends only on k, the C-dependent term
The relevant feature of this is that, since N
λ
e which
e i λ i = 0. Therefore, it does not contribute to hSi
is an odd function of q i and hqi N
λ
is, then, still given by (3.4). For later convenience, we record the analogue of (3.5),
#
"
e(q)
e
P
2d
h
Si
i
λ
e q) = −
e
S(k,
+i
.
(5.11)
C qi N
λ
e
k2 + q 2 hGi
A(d − 1)(d − 2)
The (i, j) Components of (5.8) are not affected by the C dependent terms. Hence they
⊥ in terms of S. The solution (5.11) for S then gives
yield again the solution (3.9) for Hij
e ⊥j i and hq i q j H
e ⊥ij i as in (3.10).
the same expressions for hH
j
Manipulations of the surface equation also lead to the same result as the C = 0 case.
Hence hSi and s are still related by (3.13) and finally the solution for s is given by (3.15).
The (µ, i) Components of (5.8) contain a C-dependent term and give,
µ i
e
e q) − 2 i C P kλ N
e iµ .
e ⊥µi (k, q) = − d − 2 k q S(k,
H
λ
2
2
2
2
d k +q
Ak +q
(5.12)
e ⊥µi i is still given by (3.7). Hence going through the same steps as before,
However, hqi H
we get the same solution (3.14) for Fµ .
The (µ, ν) Components of the equation of motion in momentum space is,
d−2
A
2
2
2
2 e⊥
e
kµ kν − ηµν (k + q ) S
(k + q )Hµν +
2
2
B
1
1e
2
2e⊥
i
e
e
+P (q)
kµ kν − k ηµν se − iCqi Nµν (k) − Tµν = 0 . (5.13)
k hµν +
2
2
2
⊥ and evaluate hH
e ⊥ iµν , the C-dependent term drops out (because
When we solve this for Hµν
of integration over an odd function of q i ). Using this to compute e
h⊥
µν again gives the same
result (3.16) as the C = 0 case.
The y i Equation of motion (5.9) follows also as a consequence of the metric equation:
e ⊥ and (3.9) to eliminate H
e ⊥ in favour of
e ⊥ = qiqj H
contract (5.13) with kµ kν , use kµ kν H
µν
ij
ij
e This gives (5.9) in momentum space.
S.
Thus, this constraint does not affect the solutions for the brane fields, but it provides
⊥ = ∂ i ∂ j H ⊥ and (3.9) to
an equation for F i as follows: Using Hλ⊥λ = −Hi⊥i , ∂ µ ∂ ν Hµν
ij
rewrite these in terms of S, one gets
e = 0.
e i = − i (d − 1)(d − 5) k2 hq i Si
kµ kν N
µν
2d
(5.14)
e = 0, or h∂ i Si = 0. Using
Ignoring the 4-dimensional harmonic solutions, this gives hq i Si
the solution for S in (5.10), this becomes (with no sum over j),
"Z
#
e2 (q)(q j )2 j λ P
2d
C
e
e = −i
dn q
= 0,
(5.15)
N
hq j Si
λ
A(d − 1)(d − 2)
k2 + q2
which holds only if,
e jλ (k) ≡ 3k2 Fej + 3ihqi H
e ⊥ji i − i 3 hq j H
e ⊥i i + i 6 hq j Si
e = 0.
N
i
λ
2
d
– 28 –
(5.16)
⊥ in terms of S gives
Again, expressing Hij
3k2
!
jS
e
q
d
−
2
h
i = 0.
Fe j (k) + i
2d k2 + q 2
e 2 + q 2 )i ∝ N
e jλ = 0, so we finally get the F i equations,
But (5.11) implies hq j S/(k
λ
4 F j = 0 .
(5.17)
Hence, up to harmonic functions, F i = 0. This is a consequence of the reflection symmetry
of the set up in directions transverse to the brane. A background that breaks this symmetry
can lead to non-trivial F i . Note that for the DGP model with d = 5, the constraint equation
(5.14) is trivial and does not imply an equation for F 5 .
Thus the presence of the extrinsic curvature related corrections has no effect on BIG at
the linearized level, in particular not on the ghost problem. While this is discouraging, the
positive aspect is that such terms (that on general grounds are ubiquitous in all braneworld
models) do not cause large deviations from standard gravity.
6. Revisiting Fierz-Pauli Massive Gravity
In the remaining part of this paper, we turn to a comparison of brane induced gravity for
n > 2, with massive versions of Einstein-Hilbert gravity in 4 dimensions. In this section we
start with a review Fierz-Pauli massive gravity theory [29, 30] (for reviews, see [53, 54]).
The theory is then rewritten in a way that facilitates comparison with the 4-dimensional
effective action for BIG, to be derived in the next section. In fact, the construction presented
here for the FP theory enables us to make sense of the BIG effective action.
6.1 Fierz-Pauli massive gravity in gauge invariant variables
In standard 4-dimensional Einstein-Hilbert gravity, metric fluctuations hµν in a flat background can be made massive by introducing mass parameters, say, a and b. The combination a + b = 0 results in the ghost free Fierz-Pauli massive gravity theory [29, 30].
Since we are interested in the origin of ghost in brane induced gravity, we keep a and b
unconstrained and use the term “Fierz-Pauli theory” in this generalized sense.
The gauge invariance of linearized gravity, broken by the mass terms, is restored by the
Stückelberg technique. This amounts to performing a gauge transformation and retaining
the gauge parameters as new fields. The resulting action is,
Z
B
SF P = SEH [h] −
d4 x a(hµν + 2∂(µ f¯ν) )2 + b(hµµ + 2∂µ f¯µ )2 .
(6.1)
4
SEH [h] is read off from (2.6) for d = 4. The f¯µ are Stückelberg fields transforming as δf¯µ =
λµ . Combined with δhµν = −2∂(µ λν) , it keeps SF P invariant. The f¯µ can be interpreted
as the Goldstone fields corresponding to the broken symmetry [51]. This conveys a sense
of graviton masses arising due to spontaneous symmetry breaking, even in the absence of
a more detailed fundamental mechanism.
– 29 –
The theory can be rewritten in terms of h⊥ and s of (2.8) and the new gauge invariant
variables,
1
F̄µ = f¯µ + aµ + ∂µ φ .
(6.2)
2
Note that these are the same as the Fµ in (2.17) once we identify the Stückelberg field as
f¯µ = fµ + hAµ i + 12 ∂µ (hΦi), hence the similar notation. The Fierz-Pauli action (6.1) in
terms of the gauge invariant variables becomes,
Z
h
B
a
a
3
SF P = −
+ b )s + 4( + b)∂ · F̄ s
d4 x h⊥µν ( − + a )h⊥
µν + s( +
4
8
4
4
i
+ a ∂µ F̄ν − ∂ν F̄µ )2 + 4 (a + b) (∂ · F̄ )2 .
(6.3)
The equation of motion for F̄µ ,
a
a F̄µ + (a + 2b)∂µ (∂ · F̄ ) + ( + b)∂µ s = 0 ,
4
(6.4)
has the purely longitudinal solution,
F̄µ = −
a
4
+b
−1 ∂µ s .
2(a + b)
(6.5)
Then, the hµν equation (obtained after expressing variations δh⊥ and δs in terms of δh
using the projection operators in Appendix A),
∂µ ∂ν
3
a
a
4
⊥
) ( + + b )s + 2( + b)∂ · F̄ = Tµν /B ,
(6.6)
(− + a)hµν + (ηµν −
3
8
4
4
has the massive graviton solutions,
1 1
e
h⊥
Te⊥ ,
µν =
B k2 + a µν
se = −
2
3B k2 −
Te
1 a
2 a+b (a
+ 4b)
,
(6.7)
a
corresponding to masses m2h⊥ = a and m2s = − 12 a+b
(a + 4b). The full metric is obviously,
1
hµν = h⊥
µν + ηµν s + ∂µ F̄ν + ∂ν F̄µ ,
4
with the gauge choice f¯µ = 0. In the following paragraphs, we comment on the ghost free
theory before returning to the general case in the next subsection.
The massive FP gravity theory is, in general, not ghost free: A standard analysis of
′ shows
the gauge invariant amplitude (3.17) for interaction between sources Tµν and Tµν
that for a 6= 0, the s field is always a ghost15 , as shown, for example, in [29, 31, 44] or
15
The presence of the ghost for a + b 6= 0 can already be understood at the level of action (6.3) in
terms of the Stückelberg fields: The vector field F̄µ = Fµ⊥ + ∂µ φF has an unhealthy kinetic term because
(a + b)(∂µ F̄ µ )2 leads to a 4-derivative term for the longitudinal mode φF , resulting in ghost non-decoupling.
For a + b = 0, the 4-derivative term is eliminated. This provides an intuitive way of detecting the massive
gravity ghost.
– 30 –
more explicitly in [45]. The only way to get rid of the ghost is to set a + b = 0 which gives
ms = ∞ and s = 0. The outcome is the ghost-free Fierz-Pauli massive gravity with16 ,
1
1
1 kµ kν
⊥
e
e
hµν =
T +
T , (a + b = 0) .
B k2 + a µν 3 a k 2
An alternative solution to the ghost problem is to regard gravity as an effective theory and
give the ghost a mass above the scale of validity of the effective theory by setting a + b = δ
where δ is positive and very small. Then the inaccessible ghost pole becomes physically
irrelevant [55].
Let us discuss an issue raised in section 4.4. While the ghost free theory has s = 0 for
a generic source, there is a caveat when Tµν corresponds to a cosmological constant: When
a+ b = 0, (6.4) implies 4 s = 0 and its solution s = c1 + c2 sharmonic replaces the expression
for s in terms of ∂ · F obtained from (6.5). In general c1,2 = 0 by boundary conditions, but
for a cosmological constant source, the hµν equation gives s = c1 = −4Λ/3aB. In section
4.4, this behaviour was contrasted to that of brane induced gravity. For generic a and b
the two theories respond to the cosmological constant in the same way.
6.2 A 1-parameter family of actions for FP gravity
From the outset it was clear that massive gravitons and ghosts were common features in
both Fierz-Pauli theory (for a + b 6= 0) and BIG (for n > 2) at the linearized level. Since
FP gravity is simpler, it is natural to use it as a prototype to model some aspects of BIG,
as in [38, 55]. But how much can the similarities be stretched? To quantify this, below we
rewrite the Fierz-Pauli action in a way that helps us compare with, and make sense of, the
BIG 4-dimensional effective action, to be derived in the next section.
As far as classical solutions are concerned, the action (6.3) is not unique. To find other
equivalent actions, use the relation between s and ∂ · F̄ obtained from (6.5) to convert a
part of the s2 term into (∂ · F̄ )2 and s∂ · F̄ terms. This gives,
Z
a
h
3
B
+ b + w̄1 ss
d4 x h⊥µν (− + a)h⊥
SF P = −
µν + ss +
4
8
4
i
a
2
+ 4( + b + w̄2 )∂ · F̄ s + a ∂µ F̄ν − ∂ν F̄µ ) + 4 (a + b + w̄3 ) (∂µ F̄ µ )2 , (6.8)
4
where, w̄1 s2 + 4w̄2 ∂ · F̄ s + 4w̄3 (∂ · F̄ )2 = 0 should hold on the classical solution. The w̄’s
are further constrained by requiring that the above action leads to the correct equations
of motion for F̄µ and hµν . The outcome is,
w̄3 =
a+b
w̄2 ,
a/4 + b
w̄2 =
16
a+b
w̄1 .
a/4 + b
(6.9)
The origin of vDVZ [31, 32, 33] discontinuity, that arises in the massless limit of ghost-free FP theory,
is manifest in this formulation: Sending mh⊥ → 0, keeping ms = ∞ fixed, does not recover the standard
massless solutions (3.18) as s remains zero and the theory does not couple to Tµµ . The difference between
the two massless theories does not affect light for which Tµµ = 0. But matter is affected differently and the
definitions of the Newton constant in the two theories differ by a factor of 3/4. This shows up in comparing
the bending of light in the two cases. The discontinuity is avoided in the non-linear theory through the
Vainshtein effect [39, 40].
– 31 –
Thus we get a 1-parameter family of actions, in terms of gauge invariant variables, producing the same equation of motion as the Fierz-Pauli theory.
A point to note is that, except for w̄1 = 0, these equivalent actions will look non-local
when expressed in terms of hµν , with −1 -type non-localities arising from the projection
operators (A.1) in Appendix A.
The particular choice w̄3 = −(a + b) (implying w̄2 = −(a/4 + b)) decouples gravity
from F̄µ which become a free Abelian gauge field. The resulting action involves only h⊥
and s (corresponding to integrating out F̄µ ),
Z
h
B
a(a + 4b) i
3
4
⊥µν
⊥
SF P = −
d x h
(− + a)hµν + s +
s .
(6.10)
4
8
2(a + b)
Note that, although the solutions (6.7) can be readily obtained from here, this choice of
w̄3 is a singular limit of (6.8) in the sense that ∂ · F becomes undetermined 17 .
When re-expressed in terms of hµν , using projection operators of Appendix A, this
action remains manifestly gauge invariant, with no dependence on Stückelberg fields and
no sign of symmetry breaking. But, as a trade off, it will contain −1 -type non-localities
arising from projection operators. In this example, we know that these non-localities arise
as a result of integrating out the F̄µ and indicate the existence of a local formulation ((6.1
or (6.3)) in which Stückelberg fields are needed to restore gauge invariance. The two cases
are connected by the interpolating action (6.8). This construction is useful for obtaining
the 4-d effective action in brane induced gravity where the above steps are traversed in the
reverse order.
7. 4-d Effective Action in BIG and Broken Gauge Invariance
In order to better understand the 4-dimensional structure of BIG, we need an effective
4-dimensional description of it. In this section we work out such an effective action by
integrating out parts of the fields with support away form the brane as well as the extra
components of the metric. Then we construct an action containing the Stückelberg-like
fields Fµ , based on our understanding of massive gravity in the previous section.
This has implications for the origin of ghost and for the realization of the brane general
covariance as a spontaneously broken gauge symmetry: The Fµ or fµ (2.17) of BIG are
closely related to Stückelberg fields f¯µ in Fierz-Pauli theory (more precisely, f¯µ ∼ fµ +
hAµ i + 12 ∂µ (hΦi)). Now, f¯µ restore gauge invariances broken by the FP mass terms and,
hence, can be interpreted as the Goldstone fields associated with the broken symmetry
[51]. The appearance of their analogues fµ in BIG is thus a manifestation of spontaneous
breakdown of 4-dimensional general covariance by the bulk-brane setup. In this sense, BIG
provides a realization of the gravitational Higgs mechanism.
7.1 4 -dimensional effective action in Brane Induced Gravity
In this subsection we derive a 4-dimensional effective action for BIG by integrating out all
bulk related modes in the A and B terms of (2.1). The bulk field HM N (x|| , x⊥ − y0 ) has
17
This is obvious from the Fµ equation of motion for (6.8) which gives, 2(a+b+ w̄3 )∂ ·F = −(a/4+b+ w̄2 )s
– 32 –
a part hH ⊥ iM N (x|| ), given by (2.10), that has support only on the brane. The remaining
part ∆HM N (x|| , x⊥ − y0 ) has no support on the brane, i.e., h∆HM N i = 0, and is given by
the decomposition,
P (x⊥ − y0 )
HM N = ∆HM N +
hHiM N ,
(7.1)
Z
R
R
where, Z = dn x⊥ P 2 = (2π)n dn q Pe 2 . In terms of momentum space variables,
e
e M N (k, q) = ∆H
e M N (k, q) + P (q) hHi
e M N (k) .
H
Z
(7.2)
To determine the effective action on the brane, we eliminate, first, ∆HM N by using its
equation of motion and then solve for hHiij and hHiµj using the corresponding equations.
This leaves us with an effective action for hHiµν or, equivalently, an action for the 4 µ
dimensional brane variables h⊥
µν , s, and F .
To integrate out ∆HM N , start with the bulk action (2.6), with HM N given by the above
decomposition, and a Lagrange multiplier term implementing the condition h∆HiM N = 0,
Z
bulk
SEH + dd x(P ∆HM N ) LM N (x|| ).
(7.3)
To insure gauge invariance, LM N must satisfy ∂M (P LM N ) = 0. The momentum space
equation of motion for the variation δHM N = δ(∆HM N ) (with no brane support) is,
A 2 e ⊥M N d − 2 M N
MN 2 e
eM N (k)Pe(q) = 0 .
(p p − η
p )S − L
(7.4)
p H
+
2
d
e ⊥M N in terms of L
e = L
e M and L
e M N . Blurring
This can be easily solved for Se and H
M
M
N
e and L
e
e and hH
e ⊥ iM N .
the solutions using (2.11) then determines L
in terms of hSi
e ⊥M N gives (3.5) and ,
Substituting back in the solutions for Se and H
!
#
"
MN
M pN
e
he
g
i
P
1
d
−
2
p
⊥M
N
⊥
M
N
e ,
e
e i
hSi
(7.5)
H
= 2
−hH
+
+ 2
e
e
k + q 2 hGi
d
k + q2
hGi
where we have introduced (as a generalization of (3.8))
MN
he
gi
n
= (2π)
Z
dn q
pM pN Pe2 (q)
.
(k2 + q 2 )2
(7.6)
bulk gives a 4-dimensional action for hH
e ⊥ iM N and hSi
e (the
Substituting this back in SEH
contribution of the Lagrange multiplier term vanishes),
Z
1 h e ⊥ MN e ⊥ ∗
(d − 2)(d − 1) e e ∗
bulk,ef f
4A
SEH
= (2π)
hSihSi
d4 k
hH i
hH iM N −
e
4
d2
hGi
g iM N e ⊥ M N e ∗ i
d − 2 he
−
hH i
hSi . (7.7)
e
d
hGi
e ⊥ iij and hH
e ⊥ iiµ by expressing them in terms of hSi
e
From this we can further eliminate hH
using the solutions of the (i, j) and (µ, j) components of (3.2) along with (3.5). Then, we
– 33 –
e ⊥ iµν and hSi
e in terms of e
express hH
h⊥
e and Feµ using the surface equation (2.16) and its
µν , s
consequences. Note that the equations used so far also determine Feµ in terms of se (3.14),
brane [h] leads to the 4 -dimensional
but leave se and e
h⊥ undetermined. Finally, adding SEH
effective action,
Z
B
A d − 2 e −1 ∗
A e −1 e ⊥µν e ⊥∗ 3 2
Sef f [h] = (2π)4
k +
− k2 + hGi
se se .
h
hGi
hµν +
d4 k
4
B
8
2B d − 5
(7.8)
This gives the correct equation of motion with the correct solutions (3.15),(3.16). However,
it is not yet the final form of the effective action, as explained below.
The above action is the analogue of the Fierz-Pauli action in the form (6.10) which, as
argued there, is not its most natural (i.e., most local) form. In fact, the h⊥ -terms in the
two actions can be mapped by replacing the FP mass parameter a as,
a→e
a(k) = −
A e
hGi(k) .
B
(7.9)
Naively, the s-terms too can be mapped by replacing the FP b parameter by e
a(k)/(d − 6).
But that is not the correct comparison since it does not map the F̄µ solution of FP theory
(6.5) to the Fµ solution of BIG (3.14) (simply because the quantity U does not appear in
the map). In spite of this, the similarity between the BIG action (7.8) and the FP action
in the form (6.10), teaches us the following:
Recall that in the FP case, (6.10) was a singular limit of a 1-parameter family of actions
(6.8), a generic member of which contained the FP Stückelberg fields F̄µ . The w̄1 = 0
member of this family was the original FP action (6.1) which was local when written in
terms of hµν and f¯µ . However, the specific w̄1 choice that led to (6.10) decoupled F̄µ ,
so the corresponding solution could not be reproduced. Also this action had 1/k2 -type
non-localities when expressed in terms of hµν .
To compare, this also is the structure of the BIG effective action (7.8). It too does
not contain the worldvolume fields Fµ and, when expressed in terms of hµν , will contains
1/k2 -type non-localities arising from the projection operators (A.1) 18 . We take this as
an indication that (7.8), too, is the singular limit of a 1-parameter family of actions that
generically contain the Fµ , beside the metric.
The fact that the replacements for a and b described above cannot map the F̄µ solution
to the Fµ solution means that the 1-parameter family to which the BIG effective action
(7.8) belongs cannot be obtained from the corresponding construct (6.8) in FP theory. In
this sense, the two 1-parameter families are not equivalent. The explicit construction of
the BIG 1-parameter family is performed below and is then compared to the FP theory
for a different choice of b.
7.2 A 1-parameter family of 4-d BIG effective actions
To reinstate the Fµ into the BIG effective action, let’s get back to equation (7.7). We again
eliminate the bulk tensors in favour of h⊥ , s and Fµ , but now retain Fµ . Since the relation
e But these
Of course, the action (7.8) also contains non-localities through the k-dependence of k2 hGi.
arise as a result of integrating out bulk modes and are not related to Stückelberg fields.
18
– 34 –
between s and Fµ is already implicit in the equations employed, the outcome is not unique
and depends on the steps followed. Depending on the details, the resulting action may not
even lead to the correct equations of motion for hµν and Fµ . But this can be corrected.
Indeed, a candidate action that one directly obtains in this way is 19 ,
Z
h 1
U1 ∗
A
bulk,ef f
e
h⊥µν e
h⊥∗
ese
SEH
= (2π)4
d4 k
µν + e s
e
4
hGi
hGi
i
2 2 e2
U2 e∗
+
(7.10)
k F + (k · Fe)2 + i
k · F se − k · Fe se∗ ,
e
e
hGi
hGi
with,
"
#
1
(d − 2)2
he
gi
d−1
U1 =
− 3 (d − 7)
2(d − 4)(2d − 5) − 9
,
e
16 (d − 1)2 (d − 5)2
d−2
hGi
"
#
he
gi
1
(d − 2)
2(d − 4) − 3 − 3
U2 =
.
e
4 (d − 1)(d − 5)
hGi
(7.11)
brane [h]), indicating
This does not reproduce the correct equations of motion (after adding SEH
that the ambiguity between s and ∂ · F terms has not yet been properly resolved. To cure
this, one can use the relation between s and ∂ · F in (3.14) to write an appropriate part of
the se∗ se term in terms of k · Fe. As a shortcut, a simple inspection shows that the correct
solution Feµ = −iU (kµ /4k2 )e
s can be obtained if U2 is, somehow, replaced by U given by
(3.14). Hence, the trick is to set U2 = U + δU2 and then convert the coefficient of the
δU2 piece to se se∗ form using (3.14). This leads to an effective action containing Fµ that
reproduces the correct equations of motion,
Z
h 1
3 d−2
1
2
4
⊥µν e ⊥∗
brane
4A
e
+ U sese∗
d k
h
hµν +
Sef f [h, F ] = SEH [h] + (2π)
e
e
4
4 d−5
hGi
4hGi
i
2 2 e2
U e∗
+
(7.12)
k F + (k · Fe)2 + i
k · F se − k · Fese∗ .
e
e
hGi
hGi
But, as experience with FP theory shows, this action may not be unique in this respect.
Consider converting a part of the ss∗ term into (∂F )2 and ∂F s terms, thereby, shifting the
corresponding coefficients by ω1 , ω2 and ω3 to (for convenience we use the notation (7.9)),
h
R
3 2
3 d−2
U2
a)e
h⊥µν e
h⊥∗
a( 16
+
+
w
)
sese∗
Sef f [h, F ] = −(2π)4 B4 d4 k (k2 + e
1
µν − 8 k − e
d−5
4
i
a(kµ Feν − kν Feµ )2 + e
+i (U + w2 )e
a k · Fe∗ se − k · Fese∗ + e
a(4 + w3 )(k · Fe)2 . (7.13)
Clearly, the modification is constrained by w1 sese∗ + i w2 (k · Fe∗ se − k · Fese∗ ) + w3 (k · Fe)2 = 0.
Then the equations of motion reproduce the correct solutions provided,
w3 =
4
w2 ,
U
w2 =
4
w1 .
U
(7.14)
e ⊥ iµν hSi
e using the surface equation (2.16). Instead, writing kµ kν hH
e ⊥ iµν =
We have rewritten kµ kν hH
⊥ij
e
hqi qj H i and using the second equation in (3.10) would have given a different result.
19
– 35 –
Hence, we have a 1-parameter family of correct BIG effective actions.
The Fµ equation of motion,
2k2 Feµ + (2 + w3 ) kµ (k · Fe ) + i(U + w2 )kµ se = 0 ,
gives the correct Fµ solution (3.14), except for w2 = −U (implying w3 = −4) which leads
again to the action (7.8). For this special value, ∂ µ (∂µ Fν − ∂ν Fµ ) = 0 and Fµ is pure gauge,
but otherwise undetermined. Hence, as expected, (7.8) arises as the singular limit of the
family (7.13). The the complete effective action, capable of determining all worldvolume
fields h⊥ , s and F , should be a non-singular element of this family.
The above discussion also highlights a structural similarity between the BIG and FP
actions in the forms (7.13) and (6.8) which are inequivalent otherwise: The specific choices
of the parameters wi and w̄i that decouple Fµ and F̄µ , respectively, from gravity, also
convert the F ’s into free Maxwell fields. This, in particular, decouples the longitudinal
mode of the vector field which is ghost-like, thereby, transferring the ghost entirely to
the gravity sector. In this respect, the ghosts in BIG (for n > 1) and in FP theory (for
a + b 6= 0) have similar origins, both being related to the survival of the gauge dependent
components of the brane metric (contained in Fµ ).
7.3 Spontaneous breaking of 4-d gauge invariance
The discussion in the previous section shows that the correct BIG effective action is a nonsingular member of the family (7.13). The presence of Fµ in this action has the outcome that
gauge dependent components of the metric, aµ and φ (2.8), do not drop out of the action and
gauge invariance is maintained only through the compensating transformations of fµ . Thus,
the 4-dimensional gauge invariance of the theory, i.e., symmetry under δhµν = −2∂(µ λν) ,
which is manifest in the starting BIG action, is broken spontaneously in the effective 4dimensional theory, as signaled by the need for compensating transformations.
This effect can be traced to the fact that the process of integrating out non-brane
components of graviton, by partially solving the equations of motion, requires eliminating
gauge degrees of freedom through either explicit gauge fixing, or the use of gauge invariant
variables. In our approach, this is technically related to the necessity of using (2.16) in
relating gauge invariant bulk and brane variables. In other approaches, the same issue will
arise though it may be obscured by an explicit gauge fixing.
To make the spontaneously broken nature of the symmetry very explicit, we can write
the effective action in terms of the gauge non-invariant variables hµν and fµ . A familiar
form is obtained by matching some terms of the BIG effective action (7.13) (by choosing
an appropriate w1 ) to the starting form of the FP action (6.3). For example, we can map
the coefficients of the s2 and (∂F )s terms to FP theory by choosing
w1 = −
Then, in terms of
A e
e
a(k) = − hGi(k)
,
B
U
U2
3 d−2
(
)−
.
16 d − 5 U + 2
4
e
a
e
b(k) =
4
– 36 –
3 d−2 1
−1 ,
2 d−5 U +2
the BIG effective action takes a form very similar to the FP action, differing from it only
in the (∂F )2 term. Finally, in terms hµν and fµ , it becomes,
"
Z
2
2
B
µ
4
Sef f [h, f ] = SEH [h] −
hµ + 2i kµ feµ d k e
a(k)e
hµν + 2i k(µ feν) + eb(k)e
4
#
2
µ kν
k
2
1
2U
+
1
hµν . (7.15)
h−
−2eb(k)
k · fe + e
U
6
3 k2
The first line is an exact map from the FP theory. The second line shows the structural
difference between massive FP gravity and the BIG effective action for n > 1.
Finally, one may comment on the similarity of brane induced gravity with the gravitational Higgs mechanism of ([35, 36]). In these works, in order to break general covariance,
one introduces 4 scalars fields φµ (σ) = σ µ +δφµ , with a Minkowski metric on the field space
(σ µ denote 4 dimensional spacetime coordinates). These fields are very similar to our brane
embedding coordinates, xµ (σ) = σ µ + f µ which naturally come with a Minkowski metric
(the transverse fluctuations y i are not relevant). In this sense, BIG provides a natural
realization of Higgs mechanism in gravity.
Note that the presence of extra dimensions is crucial for this similarity to hold. The
reason is that, as discussed earlier, the f µ are non-trivial only for n > 0. It is interesting
to explore the connection between these two theories further.
8. Conclusions
Our results are summarized and discussed in detail in section 1. Here we summarize
the main conclusions. The existing classical solutions for bulk gravity sourced by a 3brane tension are not adequate for studying metric fluctuations on the brane. Hence we
use the flat background approximation but argue that the results also shed light on the
curved background analysis. In the flat approximation, brane induced gravity filters out
a cosmological constant using the mass of the unhealthy scalar mode. This mass is partly
an artifact of the flat approximation and there is no real tachyon instability (in the sense
of an unbounded Hamiltonian) associated with it.
The presence of ghost in BIG is related to the phenomenologically interesting values
of parameters in the theory, corresponding to a very low graviton mass and effectively
4-dimensional gravity on the brane (for example, there is no problem with the natural
parameter values in the string theory D-brane setups). For these parameter values avoiding
the ghost is difficult even in a curved background sourced by Λ since our analysis can already
probe such backgrounds for small values of Λ and the problem is very similar to ghosts in
ordinary massive gravity. If this turns out to be the case, then BIG cannot be regarded as
a viable solution to the cosmological constant problem.
The only healthy exception would be a background that absorbs the cosmological
constant by curving directions transverse to the brane while leading to massless gravity
on the brane (Σcurved (0) = 0), if that is possible at all. This would avoid the ghost issue
and lead to a theory with filtered out Λ and modified massless gravity on the brane. The
– 37 –
modifications to gravity can be computed using the methods of section 3.3 and will have
observable consequences. However, verifying this possibility requires constructing brane
classical solutions that satisfy the equations of motion with the correct source terms. As
indicated earlier, finding such solutions is a generic problem in brane physics. At the
moment, we cannot say more on this possibility (however, see see footnote 3).
Another conclusion is that, using gauge invariant variables, it is possible to keep track
of symmetries and explicitly show that, in the effective 4-dimensional gravity theory, gauge
dependent components of the metric do not decouple. This indicates a spontaneous breaking of 4-dimensional general covariance as a consequence of the bulk-brane setup. This
setup generates the extra gauge invariant modes that contain the Stückelberg fields associated with the spontaneous breakdown of symmetry. It provides a realization of the
gravitational Higgs mechanism.
Acknowledgments
We would like to thank I. Antoniadis, M. Baumgartl, F. Berkhahn, C. Burgess, M. Berg, C.
Deffayet, J. Enander, C. de Rahm, G. Gabadadze, M. R. Garousi, C. Germani, A. Ghodsi,
H.T. Hansson, K. Hinterbichler, J. Khoury, E. Kiritsis, P. Moyasseri, S. Ramgoolam, R.
Rosen, B. Sundborg, A. Tolley, for useful discussions and comments.
A. Projection operators for gauge invariant variables
′
M ′
For a symmetric tensor HM N , its transverse part HM
N satisfying ∂ HM N = 0 is given by
−1
−1
′
L
L
L K
HM
N = HM N − d (∂M ∂ HLN + ∂N ∂ HM L − d ∂M ∂N ∂ ∂ HLK )
Then, the components in the decomposition (2.5) are given by (with H = HLL ),
∂M ∂N
1
(ηM N −
)H ′ ,
d−1
d
1 L
∂ L ∂ K HLK
=
(∂ HLN − ∂N
),
d
d
⊥
′
HM
N = HM N −
AN
d
∂ L ∂ K HLK
(H −
),
d−1
d
d ∂ L ∂ K HLK
1 H
Φ=
(
)−
d−1
d d
d − 1 d
S=
(A.1)
The gauge transformation δHM N = −2∂(M ξN ) results from the variation of AM + 12 ∂M Φ.
⊥
M = 0.
HM
N and S are invariant except under a restricted class of transformation with d ξ
These become relevant when d HM N = 0 which is not the case in brane induced gravity.
The energy-momentum tensor TM N can be decomposed in a similar way,
1
⊥
TM N = TM
N + ∂M TN + ∂N TM + ∂M ∂N Tφ + ηM N Ts
d
The components in this expansion are given by expressions analogous to those for HM N in
′
terms of TM N and a TM
N.
′
Generically, d TM N 6= 0. Then ∂ M TM N = 0 implies TM
N = TM N and other sim−1 L K
plifications. However, if d TM N = 0, then d ∂ ∂ TLK 6= 0 and one should not use
Λ
the simplified equations. An example is the cosmological constant source TM
N = ηM N Λ.
Then, the simplified equations do not hold, while using the complete projection operators
Λ⊥ = T Λ = T Λ = 0 and T Λ = dΛ.
one gets, TM
s
N
M
φ
– 38 –
References
[1] G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B 485 (2000) 208
[arXiv:hep-th/0005016].
[2] G. R. Dvali and G. Gabadadze, Phys. Rev. D 63, 065007 (2001) [arXiv:hep-th/0008054].
[3] G. Gabadadze, arXiv:hep-th/0408118.
[4] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1.
[5] G. Dvali, G. Gabadadze and M. Shifman, Phys. Rev. D 67, 044020 (2003)
[arXiv:hep-th/0202174].
[6] G. Dvali, G. Gabadadze and M. Shifman, arXiv:hep-th/0208096.
[7] R. Bousso and J. Polchinski, JHEP 0006 (2000) 006 [arXiv:hep-th/0004134].
[8] D. A. Demir, Found. Phys. 39 (2009) 1407 [arXiv:0910.2730 [hep-th]].
[9] J. Polchinski, arXiv:hep-th/0603249.
[10] S. Nobbenhuis, Found. Phys. 36 (2006) 613 [arXiv:gr-qc/0411093].
[11] S. Nobbenhuis, arXiv:gr-qc/0609011.
[12] C. de Rham, S. Hofmann, J. Khoury and A. J. Tolley, JCAP 0802 (2008) 011
[arXiv:0712.2821 [hep-th]].
[13] C. de Rham, J. Khoury and A. J. Tolley, Phys. Rev. D 81 (2010) 124027 [arXiv:1002.1075
[hep-th]].
[14] R. Gregory, Nucl. Phys. B 467 (1996) 159 [arXiv:hep-th/9510202].
[15] C. Charmousis, R. Emparan and R. Gregory, JHEP 0105 (2001) 026
[arXiv:hep-th/0101198].
[16] V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 125 (1983) 139.
[17] G. Dvali, G. Gabadadze, X. r. Hou and E. Sefusatti, Phys. Rev. D 67, 044019 (2003)
[arXiv:hep-th/0111266].
[18] S. Corley, D. A. Lowe and S. Ramgoolam, JHEP 0107 (2001) 030 [arXiv:hep-th/0106067].
[19] F. Ardalan, H. Arfaei, M. R. Garousi and A. Ghodsi, Int. J. Mod. Phys. A 18, 1051 (2003)
[arXiv:hep-th/0204117].
[20] I. Antoniadis, R. Minasian and P. Vanhove, Nucl. Phys. B 648, 69 (2003)
[arXiv:hep-th/0209030].
[21] E. Kohlprath and P. Vanhove, arXiv:hep-th/0409197.
[22] F. T. J. Epple, JHEP 0409 (2004) 021 [arXiv:hep-th/0408105].
[23] E. Kiritsis, N. Tetradis and T. N. Tomaras, JHEP 0108, 012 (2001) [arXiv:hep-th/0106050].
[24] O. Corradini, A. Iglesias, Z. Kakushadze and P. Langfelder, Phys. Lett. B 521 (2001) 96
[arXiv:hep-th/0108055].
[25] N. Kaloper and D. Kiley, JHEP 0705 (2007) 045 [arXiv:hep-th/0703190].
[26] N. Kaloper, Mod. Phys. Lett. A 23 (2008) 781 [arXiv:0711.3210 [hep-th]].
– 39 –
[27] G. Gabadadze and M. Shifman, Phys. Rev. D 69, 124032 (2004) [arXiv:hep-th/0312289].
[28] S. L. Dubovsky and V. A. Rubakov, Phys. Rev. D 67, 104014 (2003)
[arXiv:hep-th/0212222].
[29] M. Fierz, Helv. Phys. Acta 12 (1939) 3.
[30] M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173 (1939) 211.
[31] H. van Dam and M. J. G. Veltman, Nucl. Phys. B 22 (1970) 397.
[32] V. I. Zakharov, JETP Lett. 12 (1970) 312 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 447].
[33] Y. Iwasaki, Phys. Rev. D 2 (1970) 2255.
[34] M. Kolanovic, M. Porrati and J. W. Rombouts, Phys. Rev. D 68 (2003) 064018
[arXiv:hep-th/0304148].
[35] G. ’t Hooft, arXiv:0708.3184 [hep-th].
[36] Z. Kakushadze, Phys. Rev. D 77, 024001 (2008) [arXiv:0710.1061 [hep-th]].
[37] N. Arkani-Hamed, S. Dimopoulos, G. Dvali and G. Gabadadze, arXiv:hep-th/0209227.
[38] G. Dvali, S. Hofmann and J. Khoury, Phys. Rev. D 76 (2007) 084006
[arXiv:hep-th/0703027].
[39] A. I. Vainshtein, Phys. Lett. B 39 (1972) 393.
[40] C. Deffayet, G. R. Dvali, G. Gabadadze and A. I. Vainshtein, Phys. Rev. D 65 (2002)
044026 [arXiv:hep-th/0106001].
[41] S. Weinberg, “Gravitation and Cosmology: Principles and Applications of the General
Theory of Relativity”, J. Wiley and Sons (1972) 657 p.
[42] S. L. Adler, Rev. Mod. Phys. 54 (1982) 729 [Erratum-ibid. 55 (1983) 837].
[43] I. R. Klebanov and L. Thorlacius, Phys. Lett. B 371 (1996) 51 [arXiv:hep-th/9510200].
[44] P. Van Nieuwenhuizen, Nucl. Phys. B 60 (1973) 478.
[45] F. C. P. Nunes and G. O. Pires, Phys. Lett. B 301 (1993) 339.
[46] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions”, Dover
Publications (1965) 1046 p
[47] M. E. Peskin and D. V. Schroeder, “An Introduction To Quantum Field Theory,” Reading,
USA: Addison-Wesley (1995) 842 p
[48] L. S. Brown, “Quantum Field Theory,” Cambridge, UK: Univ. Pr. (1992) 542 p
[49] H. Gustafsson, U. Lindstrom, P. Saltsidis, B. Sundborg and R. van Unge, Nucl. Phys. B
440 (1995) 495 [arXiv:hep-th/9410143].
[50] P. L. Bozhilov, arXiv:hep-th/0011032.
[51] N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Annals Phys. 305 (2003) 96
[arXiv:hep-th/0210184].
[52] Y. K. Cheung, M. Laidlaw and K. Savvidy, JHEP 0412 (2004) 028 [arXiv:hep-th/0406245].
[53] V. A. Rubakov and P. G. Tinyakov, Phys. Usp. 51 (2008) 759 [arXiv:0802.4379 [hep-th]].
[54] M. V. Bebronne, arXiv:0910.4066 [gr-qc].
[55] G. Dvali, O. Pujolas and M. Redi, Phys. Rev. Lett. 101 (2008) 171303 [arXiv:0806.3762
[hep-th]].
– 40 –
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement