On "New Massive" 4D Gravity

Abstract

We construct a four-dimensional (4D) gauge theory that propagates, unitarily,
the five polarization modes of a massive spin-2 particle. These modes are
described by a "dual" graviton gauge potential and the Lagrangian is 4th-order
in derivatives. As the construction mimics that of 3D "new massive gravity", we
call this 4D model (linearized) "new massive dual gravity". We analyse its
massless limit, and discuss similarities to the Eddington-Schroedinger model.

Full text

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arXiv:1202.1501v1 [hep-th] 7 Feb 2012
UG-12-09
DAMTP-2012-10
On “New Massive” 4D Gravity
Eric A. Bergshoeff1, J.J. Fern´ andez-Melgarejo1,2, Jan Rosseel1
and Paul K. Townsend3
1Centre for Theoretical Physics, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
email: E.A.Bergshoeff@rug.nl, j.rosseel@rug.nl
2Grupo de F´ ısica Te´ orica y Cosmolog´ ıa,
Dept. de F´ ısica, University of Murcia,
Campus de Espinardo, E-30100 Murcia, Spain
email: jj.fernandezmelgarejo@um.es
3Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, U.K.
email: P.K.Townsend@damtp.cam.ac.uk
ABSTRACT
We construct a four-dimensional (4D) gauge theory that propagates, unitarily, the
five polarization modes of a massive spin-2 particle. These modes are described by
a “dual” graviton gauge potential and the Lagrangian is 4th-order in derivatives. As
the construction mimics that of 3D “new massive gravity”, we call this 4D model
(linearized) “new massive dual gravity”. We analyse its massless limit, and discuss
similarities to the Eddington-Schr¨ odinger model.

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1 Introduction
Einstein’s theory of gravity can be viewed as an interacting theory of massless spin-
2 particles, gravitons, but it is not excluded that the graviton could have a small
mass [1]. This possibility has implications for cosmology that have motivated many
recent models of “massive gravity”; see [2] for a recent review. At the linearized level,
the introduction of a mass for the spin-2 field is straightforward, and it leads to the
well-known Fierz-Pauli (FP) theory. The difficulty is to find a consistent interacting
version of this theory.
This difficulty is much less severe in three spacetime dimensions (3D), as is shown
by “new massive gravity” (NMG) [3] in which a mass scale is associated with curvature-
squared terms in the action. Despite being “higher-derivative”, NMG has a linearized
limit that is equivalent to the ghost-free 3D FP theory, and hence propagates a parity-
doublet of massive spin-2 states. In addition, the Boulware-Deser ghosts [4] that afflict
generic interacting higher-derivative theories are absent [5].
In this paper we explain how some of the ideas underlying the success of NMG as
a 3D massive gravity theory can be extended to higher dimensions, in particular four
spacetime dimensions (4D). At first sight, this seems impossible because the addition of
higher-derivative interactions to the 4D Einstein-Hilbert action never yields a ghost-free
theory propagating massive gravitons [6]. However, this problem can be circumvented,
at least at the linearized level, by using a “dual” field representation to describe the
graviton, i.e. not the usual symmetric tensor field but some other gauge field that
describes the same (massive) degrees of freedom.
The idea that the graviton might be described by a field in a “dual” representation of
the Lorentz group is realised by the Eddington-Schr¨ odinger (ES) model [7,8], which has
an affine connection as the fundamental field. We review this model in our concluding
section because there are some similarities to the model that we propose here, and
it shows that “exotic” Lorentz representations can be compatible with interactions.
Other “exotic” Lorentz representations were explored in [9], and the topic has been
reconsidered more recently in the context of string/M-theory dualities, see, e.g. [10,
11]. However, what we need here is a dual representation for massive spin 2. In D
spacetime dimensions, the possible spin-2 field representations are those induced by
representations of SO(D−1) corresponding to two-column Young tableaux; for D ≤ 4
the correspondence is one-to-one1. The standard (symmetric tensor) representation
has one box in each of the two columns while the dual representation has (D − 2)
boxes in the first column and one in the second column. The 3D case is special in that
the “dual” field is again a symmetric tensor field2.
A Fierz-Pauli-type model for the 4D dual-field representation of a massive spin-2
particle was constructed by Curtright under the rubric of “generalized FP theory” [12]
(see also [13]). It was pointed out relatively recently [16] that Curtright’s model can be
1Any discussion of this issue for D > 4 depends on what is meant by “spin”.
2Four dimensions is similarly special for massless spin-2 duality.
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found by an application of “connection-metric duality”. Although the dual spin-2 field
is not a gauge field in this context, there is a procedure (introduced in [15] and explained
in detail in [17]) for converting 3D FP-type models into equivalent higher-derivative
gauge theories. The basic idea is to solve the differential “subsidiary condition” on
the FP field. A special feature of 3D is that the gauge field thus introduced is in the
same Lorentz representation as the original FP field, i.e. a rank-s symmetric tensor for
spin s. This allows the integration of the field equations to a (higher-derivative) gauge
invariant action, and this action is ghost-free if it preserves parity, which it does if s is
even3. Linearized NMG may be found this way by starting with the 3D FP theory for
spin 2.
A naive extension of this procedure to D > 3 fails because the gauge field found
by solving the differential constraint on the FP field is no longer in the same Lorentz
representation as the original FP field. However, if one takes the dual FP theory as
the starting point then the gauge field is in the same (dual) Lorentz representation,
so one can integrate the gauge-field equations to a (higher-derivative) action, which
is ghost-free for some choice of overall sign because all propagated modes are related
by rotational invariance. Here we use this observation to construct a fourth-order but
ghost-free 4D field theory for massive spin 2. It is a 4D analog of the linearized 3D
NMG but with a “dual” field describing the graviton, so we shall call it (linearized)
“new massive dual gravity” (NMDG). This linear massive spin-2 model is potentially
the linearized limit of an interacting NMDG theory describing 4D massive gravity but
we postpone further discussion of this issue to the conclusions.
One issue that can be addressed in the context of a linear theory, at least partially,
is the nature of the massless limit. As is well-known, this limit is singular for FP theory
because whereas the FP action describes the five polarization states of a massive spin-2
particle its massless limit coincides with the linearized Einstein-Hilbert action, which
propagates only the two modes of a massless spin-2 particle. In the case of NMDG it
is less clear what might be meant by the “massless limit”. One option is to consider
the 4th-order term by itself. The analogous massless limit of 3D NMG was analysed
by Deser [14], who found that it propagates a single mode, in contrast to NMG itself,
which propagates two modes. Here we analyse the 4D “pure” 4th-order model defined
by the 4th-order term of NMDG. We find that it propagates three modes, one scalar
and two spin-2 modes, in contrast to NMDG itself, which propagates an additional two
helicity-1 modes.
We also show that the massless limit of NMDG can be taken in another way, and
one that does not lead to any discontinuity in the number of propagated modes. In the
massless model that results from this limit, the spin-1 modes are present as Goldstone
bosons which become the helicity-1 modes of a massive graviton in NMDG itself. This
is a novel example of the St¨ uckelberg mechanism, which is essentially an affine version
of the Higgs mechanism.
3This point has been clarified in [18].
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2 Connection-Metric duality
We begin by reviewing how connection-metric duality may be used to construct a
“generalised FP” field theory for a “dual” spin-2 field on a Minkowski space-time of
arbitrary dimension D [16]. Then we focus on the D = 3,4 cases.
A convenient starting point for this construction is the first-order form of the
Einstein-Hilbert Lagrangian, deteR(ω), in which the vielbein eµaand spin-connection
ωµab= −ωµbaare independent fields. We then linearize about a Minkowski vacuum by
writing the vielbein as
eµa= δµa+ hµa. (1)
and expand the action to second order in fields. After adding a Fierz-Pauli mass term
for the perturbation h, we arrive at the Lagrangian
L = −2h(∂ · ω)+ 2hµν(∂αωνµα+ ∂νωµ) − ω · ω − ωναρωαρν−m2?hµνhνµ− h2?, (2)
where m is the mass, and
ωµ= ωααµ,h = hµµ. (3)
It should be noted that hµνis a general second-order tensor. For m2= 0 the action is
invariant under the gauge transformations
δhµν= ∂µξν+ Λµν,δωµνρ= ∂µΛνρ,(4)
where ξµand Λµν= −Λνµare the parameters of the linearized general coordinate and
Lorentz transformations, respectively.
Eliminating the spin-connection from the action gives rise to the following La-
grangian:
Lh= h(δν)εδγµενρα∂ρ∂γhαµ− m2(hµνhνµ− h2). (5)
The antisymmetric part of hµν appears only in the mass term and is therefore an
auxiliary field that can be trivially eliminated; this yields the usual spin-2 FP action
for a symmetric tensor field. Alternatively, we can eliminate from (2) the entire tensor
hµν in terms of the spin-connection. After multiplication by m2this yields the (D-
dimensional) dual Lagrangian
L(D)
dual=
?
KµνKνµ−
D
4(D − 1)K2
?
− m2(ω · ω − ωµνρωρµν) ,(6)
where
Kµν= ∂αωµνα+ ∂µων,K = Kµµ= 2∂ · ω . (7)
It is convenient to rewrite this in terms of a rank-(D −1) tensor U, defined by writing
the spin-connection in the form
ωµαβ= εαβν1···νD−2Uν1···νD−2µ.(8)
The U-tensor is antisymmetric on its first (D − 2) indices. We now discuss separately
the special cases D = 3 and D = 4.
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2.1
D = 3
In this case (8) becomes
ωµνρ= ενραUαµ, (9)
where U is a general second-rank tensor. Substitution into the kinetic term of the dual
Lagrangian (6) yields
KµνKνµ−3
8K2= Uδνεδγµενρα∂ρ∂γUαµ−1
2(εµνρ∂µUνρ)2. (10)
Using a Schouten identity, one can show that
1
2(εµνρ∂µUνρ)2= U[δν]εδγµενρα∂ρ∂γUαµ, (11)
so that
Ldual= U(δν)εδγµενρα∂ρ∂γUαµ− m2(UµνUνµ− U2),(12)
where, here, U is the trace of the matrix U. This has precisely the form of the FP
Lagrangian (5). We thus find that the dual Lagrangian is equivalent to the usual FP
Lagrangian for a massive spin-2 field.
2.2
D = 4
In this case (8) becomes
ωµνρ= ενραβUαβ,µ. (13)
For clarity, we have inserted a comma to help recall which is the antisymmetric index
pair: U(αβ),µ= 0. Substitution into the kinetic terms of the Lagrangian (6) for D = 4
yields
KµνKνµ−1
3K2= Uγδ,νενραβεγδσµ∂ρ∂σUαβ,µ−1
3
?εαβγδ∂αUβγ,δ
?2.(14)
The 4D identity analogous to the 3D identity (11) is
1
3
?εαβγδ∂αUβγ,δ
?2= U[γδ,ν]εγδσµενραβ∂ρ∂σUαβ,µ.(15)
As a consequence, the totally antisymmetric part of the U-tensor cancels from the
kinetic term and can be trivially eliminated to yield a Lagrangian in terms of the field
Tµν,ρ= Uµν,ρ− U[µν,ρ],(16)
which is a mixed-symmetry tensor with zero totally antisymmetric part. After multi-
plication by a factor of 1/4, we arrive at the dual Lagrangian
Ldual=1
4Tγδ,νενραβεγδσµ∂ρ∂σTαβ,µ−m2
2
(Tµν,ρTµν,ρ− 2TµTµ) ,(17)
where Tµ= Tµν,ρηνρ. In terms of the “generalized Einstein tensor”
Gµν,ρ(T) =1
2εµναβεργδǫ∂α∂γTδǫ,β, (18)
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the dual Lagrangian takes the form
Ldual=1
2Tµν,ρGµν,ρ(T) −1
2m2(Tµν,ρTµν,ρ− 2TµTµ) .(19)
The generalized Einstein tensor is a mixed-symmetry tensor of the same algebraic
type as T. It satisfies the Bianchi-type identity
∂µGµν,ρ≡ 0,∂ρGµν,ρ≡ 0.(20)
The first of these implies the other, as a consequence of the fact that G[µν,ρ]≡ 0. It
also implies that
∂µGµ= 0,
Gµ= Gµν,ρηνρ.(21)
Another feature of the generalized Einstein tensor is that it defines a self-adjoint ten-
sor differential operator acting on tensors of the same algebraic type as T; the field
equations for T are therefore
Gµν,ρ(T) = m2?Tµν,ρ− 2T[µην]ρ
?. (22)
An equivalent set of equations is
(✷ − m2)Tµν,ρ= 0,Tµ= 0,∂ρTρµ,ν= 0.(23)
The first of these is dynamical while the other two are “subsidiary” conditions, which
ensure that only the five polarization modes of a spin-2 particle are propagated; this
can be shown by a straightforward analysis. Also, because T[µν,ρ]≡ 0, by definition,
the differential subsidiary condition implies that
∂ρTµν,ρ= 0. (24)
3 New Massive Dual Gravity
We may find a new set of higher-order equations equivalent to the second-order equa-
tions (23) by solving the differential subsidiary condition. To do so we use the Poincar´ e
lemma for differential forms in D-dimensional Minkowski spacetime: for any p-form P,
∂µPµν1...µp−1= 0⇒Pµ1...µp= εµ1...µpν1...νqρ∂ρQν1...νq
(q = D − 1 − p)(25)
for some q-form Q, which is defined modulo a closed q-form. We can apply this to any
tensor that is divergence-free on some set of p anti-symmetrized indices to get a dual
gauge potential in which the set of p indices are replaced by a set of q indices.
In the case of the differential subsidiary condition on the T-tensor in (23) we have
to take into account that it also satisfies (24). We therefore have to use the D = 4 case
of (25) twice (once with p = 1 and once with p = 2). This leads to an expression for T
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as a second-order differential operator acting on a tensor S of the same algebraic type
as T. In fact, one finds that
Tµν,ρ= Gµν,ρ(S), (26)
where the S-tensor is defined by this equation modulo a gauge transformation of the
type
Sµν,ρ→ Sµν,ρ+ ∂ρΛµν− ∂[µΛν]ρ+ ∂[µΞν]ρ, (27)
where Λ is an antisymmetric tensor parameter and Ξ a symmetric tensor parameter.
What was the differential subsidiary condition on T has now become the Bianchi-type
identity for G(S). The construction gives us the general solution of the differential
subsidiary condition, so G(S) = 0 should imply that S is pure gauge. This will be
verified explicitly in subsection 3.3.
Using (26) in the remaining equations of (23) we find the following two gauge-
invariant equations for S:
(✷ − m2)Gµν,ρ(S) = 0,
Gµ(S) = 0. (28)
By construction, these equations are equivalent to the “generalised FP” equations
(23). They can be derived from the following “new massive dual gravity” (NMDG)
Lagrangian (which is gauge-invariant up to a total derivative):
LNMDG= −1
2Sµν,ρGµν,ρ(S) +
1
2m2Sµν,ρCµν,ρ(S), (29)
where
Cµν,ρ= ✷Gµν,ρ− ∂ρ∂[µGν]+ ηρ[µ✷Gν]. (30)
The C-tensor is of the same algebraic type as the generalized Einstein tensor, and
hence of the new mixed-symmetry tensor field S, except that it is traceless. It also
satisfies the same Bianchi identities as the generalized Einstein tensor. Furthermore,
it defines a self-adjoint tensor operator acting on tensors of the same algebraic type as
S. Using this last property, we deduce that the NMDG field equation is
m2Gµν,ρ(S) − Cµν,ρ(S) = 0. (31)
Taking the trace of this equation, we deduce that Gµ(S) = 0, and hence from (30)
that Cµν,ρ = ✷Gµν,ρ. Using this in (31), we deduce that Gµν,ρis annihilated by the
Klein-Gordon operator, and hence that equation (31) is, as claimed, equivalent to the
two equations of (28).
We have now shown that the field equations of the NMDG Lagrangian (29) are
equivalent to the (4D) spin-2 FP equations, despite the fact that this Lagrangian is
4th-order in derivatives. Because all five propagating modes are in an irreducible repre-
sentation of the rotation group, they will all be propagated unitarily for an appropriate
choice of sign, which is the one we have chosen. For the remainder of this section, we
explore a few features of this 4D model that are suggested by its 3D cousin, NMG, and
we consider its infinite mass limit.
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3.1Dimensional reduction
The analogy of 4D NMDG theory with 3D NMG may be clarified by showing how the
latter follows from a truncated dimensional reduction of the former. We will split the
4D indices as
µ = {m,z}, (32)
where z denotes the compactified direction. All fields are assumed to be independent
of z. We then define
hmn= S(m|z|,n), (33)
and set all other components of S to zero. Using this reduction/truncation in the
action (29), we arrive at the 3D Lagrangian
Lh = 2hµνGµν(h) +
4
m2
?
Gµν(h)Gµν(h) −1
2G(h)G(h)
?
, (34)
where the scalar G is the 3D Minkowski trace of the symmetric tensor Gµν, which is
the linearized Einstein tensor for the symmetric tensor h. This is one form of the 3D
NMG Lagrangian for a massive spin-2 field.
3.2Auxiliary field formalism
The C tensor appearing in the NMDG Lagrangian (29) can be written in the form
Cµν,ρ= 2Gµν,ρ(S(S)) , (35)
where the S-tensor is an analog of the Schouten tensor:
Sµν,ρ= Gµν,ρ+ ηρ[µGν].(36)
Using the self-adjointness of the operator defined by the tensor G(S), we may rewrite
the Lagrangian (29) as
LNMDG= −1
2Sµν,ρGµν,ρ(S) +
1
m2Gµν,ρ(S)Sµν,ρ(S).(37)
Now consider the alternative two-derivative Lagrangian, involving an auxiliary field
f of the same algebraic type as the field S (i.e. fab,c= −fba,cand f[ab,c]= 0):
L = −1
2Sµν,ρGµν,ρ(S) +
1
m2
?
fµν,ρGµν,ρ(S) −1
2(fµν,ρfµν,ρ− 2fµfµ)
?
. (38)
The field equation for f is
fµν,ρ= Sµν,ρ(S).(39)
Using this equation to eliminate the f-field from (38) we recover the NMDG action in
the form (37).
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The alternative NMDG Lagrangian (38) allows an alternative proof of the equiva-
lence of the linearized 4D NMG to the dual FP theory. It is simplest to first diagonalize
by setting
Sµν,ρ=˜Sµν,ρ+
1
m2fµν,ρ. (40)
We then find that the Lagrangian (38) becomes
L = −1
2
˜Sµν,ρGµν,ρ(˜S) +
1
2m4
?fµν,ρGµν,ρ(f) − m2(fµν,ρfµν,ρ− 2fµfµ)?.(41)
The first term is just the infinite mass limit of the Lagrangian (38) expressed in terms of
˜S. As it propagates no modes (a statement that we verify in the subsection to follow)
it may be ignored. The remaining terms are just those of the dual FP Lagrangian
(19) expressed in terms of the f-tensor field. This argument is analogous to the one
originally used to show the equivalence of 3D NMG to the 3D FP theory [3].
3.3Infinite mass limit
In the long distance limit, which is equivalent to a limit in which the graviton mass
becomes infinite, the Lagrangian of NMDG reduces to the second-order one
L2= −1
2Sµν,ρGµν,ρ(S). (42)
The equation of motion is now
Gµν,ρ(S) = 0. (43)
Since the mass of all propagating modes has been sent to infinity, we should expect
this equation to propagate no modes. We now confirm this by an explicit canonical
analysis.
To make a time/space split we set
µ = {0,i} ,i = 1,2,3,(44)
and parametrize the components of the field Sµν,ρas follows:
Sij,k = εijlUlk+ 2δk[iζj],
S0i,j
= Wij+ εijkvk,
Sij,0 = −2εijkvk,
S0i,0 = wi.(45)
In the above decomposition, Uijis a symmetric, traceless 3-tensor, Wijis a symmetric
3-tensor and ζi, vi and wi are 3-vectors. We will fix the gauge invariance (27) by
imposing a De Donder-type gauge fixing condition
∂iSiµ,ν+1
2∂νSµi,i= 0.(46)
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Written out in terms of the fields U, W, ζ, v, w, this gauge fixing condition is given by
εimn∂mUnj= 0,
∂iζi= 0,
∂jWji−1
2∂iW = εimn∂mvn,
εimn∂mvn=1
2
˙W = 2∂iwi,
˙ζi,
(47)
with W = Wii. Note that the above gauge conditions represent 5, 1, 3, 2 and 1
independent conditions respectively. The 16 gauge transformations (27) are reducible.
Writing (27) in terms of
ξµν=1
2(Ξµν− Λµν) ,
one sees that there is a gauge invariance for the parameters ξµν
(48)
δξµν= ∂µξν+1
2∂νξµ. (49)
In total there are thus only 16 - 4 = 12 independent gauge transformations ξµν, that
are fixed by the 12 gauge fixing conditions (47).
The first condition of (47) already implies that Uijdoes not propagate any physical
modes. As
2ε[i|mn∂mUn|j]= εijm∂nUmn, (50)
the vanishing of the antisymmetric part of this gauge condition, implies that ∂jUji= 0.
Contracting the first equation of (47) with ∂kεkliand using ∂jUji= 0, leads to
∇2Uij= ∂m∂mUij= 0, (51)
and hence U does not propagate any physical degrees of freedom. We can thus set
it to zero. Similar conclusions for the other fields can be obtained by analyzing the
equations of motion (43). Upon using the gauge fixing (47), together with Uij= 0, the
components of the Einstein tensor are given by
Gij,k = δk[i¨ζj]+ 2∂[i˙Wj]k+ εijk∂m˙ vm− 3εijl∂l˙ vk
+2δk[i∇2wj]− 2∂k∂[iwj],
= −∂[i˙ζj]−1
2δij∇2W + ∇2Wij+ εijk∂k(∂pvp),
Gij,0 = 2∂[i˙ζj]− 2εijk∂k(∂pvp),
G0i,0 = ∇2ζi.
G0i,j
(52)
From G0i,0= 0, one finds that ζidoes not propagate any physical modes either and
we will thus also set ζi= 0. From Gij,0= 0, one then infers that ∂ivi= 0. Together
with ζi= 0 and the fourth condition of (47), one can then see that vialso does not
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propagate any degrees of freedom and can be set to zero as well. The symmetric part
of G0i,j= 0 then immediately implies that Wij is non-propagating and can be set to
zero. Finally, contracting Gij,k= 0 with δjkand using the last condition of (47) implies
that wialso does not propagate.
To summarise: the equations of motion of NMDG in the infinite mass limit do not
propagate any physical degrees of freedom, as expected. This result can be viewed
as a check on our earlier claim that the general solution of the subsidiary condition
∂µTµν,ρ= 0 is T = G(S), since we now see that G(S) = 0 implies that S = 0 once all
gauge invariances have been fixed.
4 Massless limit of NMDG
We have considered the infinite mass limit of NMDG in subsection 3.3. Now we consider
the opposite limit in which m2→ 0. We see from (29) that the fourth-order term
dominates as m2→ 0, but to actually take the limit we must first multiply by m2.
This gives us the “pure” fourth-order Lagrangian
Lm2=0= Gµν,ρ(S)Sµν,ρ(S). (53)
In addition to the gauge invariances (27) this action has the conformal-type gauge
invariance
Sµν,ρ→ Sµν,ρ+ ηρ[µΩν], (54)
although the choice Ωµ= ∂µφ for some scalar φ reproduces the Ξ-transformation of (27)
in the special case that Ξµν= −ηµνφ, so two 1-form parameters Ω and Ω′correspond
to the same conformal-type gauge transformation if they differ by an exact 1-form.
This conformal-type gauge invariance is new because it is broken by the “mass-term”
(42) that we have now dropped from the NMDG Lagrangian. As we shall see, the new
gauge invariance leads to a ‘disappearance’ of the helicity-1 modes from the spectrum,
and hence a discontinuity of the m → 0 limit4. A discontinuity of this sort was to be
expected because a similar discontinuity in the number of propagated modes occurs in
the massless limit of the 3D NMG, which is what one finds by an application to the
Lagrangian (53) of the reduction/truncation procedure of subsection 3.1; i.e. the “pure”
fourth-order 3D Lagrangian analysed by Deser [14]. This 3D model was analysed in a
different way in [15], and here we adapt this analysis to the 4D case.
First we rewrite the Lagrangian of (53) as a second-order Lagrangian by introducing
an auxiliary field, as explained in subsection 3.2. This gives us the new, but equivalent,
Lagrangian
L = fµν,ρGµν,ρ(S) −1
2(fµν,ρfµν,ρ− 2fµfµ) .(55)
4There is also a discontinuity in the number of propagated modes in the opposite, infinite-mass,
limit, but this can be explained as as an effect of decoupling at low energy.
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Now we use the self-adjointness of the generalized Einstein operator to rewrite this
Lagrangian as
L = Sµν,ρGµν,ρ(f) −1
2(fµν,ρfµν,ρ− 2fµfµ) .
In this form, we see that S is a Lagrange multiplier for the constraint Gµν,ρ(f) = 0,
which has the solution
(56)
fµν,ρ= ∂[µHν]ρ+ ∂ρJµν− ∂[µJν]ρ,(57)
for some symmetric tensor H and antisymmetric tensor J. Substituting this solution
for f (which is a procedure that can be justified by viewing it as integration over S in
the path-integral) one finds the new, but equivalent, action
L′= −1
2HµνGµν(H) −1
4FµνρFµνρ, (58)
where Gµν(H) is the linearized Einstein tensor
Gµν(H) = −1
2
?✷Hµν− 2∂(µHν)+ ∂µ∂νH?+1
2ηµν(✷H − ∂ρHρ) ,(59)
with Hµ= ∂νHµνand H = ηµνHµν, and Fµνρis the 3-form field-strength tensor
Fµνρ= 3∂[µJνρ]. (60)
This result shows that H and J are subject to the gauge-invariances
Hµν→ Hµν+ ∂(µξν),Jµν→ Jµν+ ∂[µζν], (61)
for arbitrary vector parameters (ξ,ζ). Notice that the f-tensor is itself invariant under
the particular gauge transformation for which ξ = −3ζ.
The essential point here is that the massless limit of (linearized) NMDG, defined by
the action (53), is ghost-free and propagates a massless scalar mode (represented by the
2-form gauge potential J) in addition to the two massless graviton modes propagated
by the linearized Einstein-Hilbert action for H, but there are no massless spin-1 modes.
We shall now verify this conclusion by a canonical analysis.
4.1Canonical analysis
We shall use again the parametrization (45) of the space/time components of Sµν,ρ.
Apart from its invariance under the gauge transformations (27), the Lagrangian (53)
is also invariant under the conformal transformations (54). We shall begin by fixing
these conformal transformations. A suitable gauge fixing condition is given by
Sµi,i= 0⇔ζi= W = 0. (62)
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As the gauge transformations (27) do not preserve this choice of conformal gauge, they
must be augmented by a compensating conformal transformation. The components of
the parameter Ωµof this compensating transformation are
Ω0 =
2
3
?˙ξii− 2∂iξ0i+ ∂iξi0
?
,
Ωi = ∂iξjj− 2∂jξij+ ∂jξji, (63)
where ξµν is defined in (48). The compensated ξ-transformations are then given ex-
plicitly on the fields Uij, Wij, ζi, vi, wiby the following expressions:
δUij
= ε(i|mn∂mξn|j)− ε(i|mn∂j)ξmn,
δζi = 0,
δWij
=
?∂0ξ(ij)− 2∂(iξ0j)+ ∂(iξj)0
= −1
2ǫijk
δwi = −(∂iξ00− 2∂0ξi0+ ∂0ξ0i) −1
?−1
3δij(∂0ξkk− 2∂kξ0k+ ∂kξk0) ,
δvk
?
∂iξj0−˙ξij
?
,
2(∂iξjj− 2∂jξij+ ∂jξji) .(64)
These gauge transformations are reducible; they still feature the gauge invariance (49)
of the parameter ξµν. Moreover, this parameter is now subject to an extra conformal
gauge transformation with scalar parameter Λ:
δξµν= ηµνΛ. (65)
In total there are now 16 − 5 = 11 independent ξ-transformations. These can be fixed
by the following 11 gauge fixing conditions
ǫimn∂mUnj
= 0,
∂iWij
= 0,
ǫimn∂mvn = 0,
∂iwi = 0.(66)
As in the infinite mass case, the first of these implies that Uijis non-propagating and
can be set to zero. The components of the C-tensor can then be calculated, subject to
the gauge conditions (62) and (66) and Uij= 0. Solving the third condition of (66) by
vi= ∂iφ,(67)
we find
Cij,k = εijk∇2✷˙φ − 3εijl∂k∂l✷˙φ + 2∂[i✷˙Wj]k+ 2δk[i✷∇2wj]− 2∂k∂[i✷wj]
+∂k∂[i∇2wj]− δk[i✷∇2wj],
= ✷∇2Wij+ εijk∂k∇2✷φ +1
2∂j∇2˙ wi,
Cij,0 = −2εijk∂k∇2✷φ + ∂[i∇2˙ wj],
1
2(∇2)2wi.
C0i,j
C0i,0 =(68)
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From C0i,0 = 0, it then follows that wi is non-propagating and can be put to zero.
From Cij,0= 0, one then gets that φ obeys the massless Klein-Gordon equation
✷φ = 0. (69)
Similarly, from the symmetric part of C0i,j= 0, one finds that Wijobeys the massless
wave equation
✷WTT
ij
= 0, (70)
where the superscript TT indicates that the symmetric tensor Wijis “transverse trace-
less”, i.e. traceless and subject to the gauge condition ∂iWij= 0, which implies that
WTT
ij
has two independent components, which are propagated as two helicity-2 modes.
Therefore, there is a total of three propagating modes, two of spin 2 propagated by
WTT
ij
and the other propagated by the scalar φ.
To show that these modes are propagated unitarily we must return to the La-
grangian (53) and express it in terms of the space/time components of S appearing in
(45). Upon imposing the gauge fixing conditions (62) and (66), we find that
Lm2=0= ∇2wi∇2wi+ 12(∇2φ)✷(∇2φ) − 2WTT
ij∇2✷WTT
ij . (71)
We thus confirm that there are three modes, propagated by φ and WTT
as ∇2is a negative definite operator, we also confirm that the kinetic terms for these
fields are positive, so that all three modes are propagated unitarily; i.e. they are not
“ghosts”.
ij. Moreover,
4.2Another massless limit
We will conclude this section by showing how the massless limit can be taken in another
way that avoids any discontinuity in the number of propagating modes. We begin by
returning to (29) and making the field redefinition
Sµν,ρ→˜Sµν,ρ= Sµνρ+ m−1ηρ[µAν]. (72)
This has no effect on the quartic term in the action because of its conformal-type gauge
invariance, so the Lagrangian is now
LNMDG= −1
2
˜Sµν,ρGµν,ρ(˜S) +
1
2m2Sµν,ρCµν,ρ(S). (73)
Although this depends on an additional field A, in comparison to the original NMDG
Lagrangian, it also has an additional gauge invariance: it is invariant under the
conformal-type transformation
Sµν,ρ→ Sµν,ρ+ ηρ[µΩν],Aµ→ Aµ− mΩµ. (74)
This additional gauge invariance allows us to set A = 0, thereby recovering the original
NMDG Lagrangian of (29). In other words, A is a “St¨ uckelberg field”.
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Although the Lagrangian (73) is equivalent to the NMDG Lagrangian (29) for non-
zero mass, it has a different zero-mass limit. To take this limit we first multiply by m2,
as before. Then, setting m = 0 and integrating by parts (omitting boundary terms)
we arrive at the new massless Lagrangian
˜
Lm2=0= −1
4FµνFµν+ Gµν,ρ(S)Sµν,ρ(S), (75)
where Fµν= 2∂[µAν]is the field-strength for A. The conformal-type invariance is not
lost in this limit because the conformal transformation of A also goes to zero. As
expected, however, there is a residual Maxwell-type invariance for A, ensuring that it
propagates only the two massless spin-1 modes that we previously lost in the massless
limit. What we have done is to take the massless limit in a way that does not change the
total number of gauge invariances, and this removes any discontinuity in the number
of propagated modes.
5 Conclusions
In this paper we have constructed a 4D analog of the linearized 3D massive gravity
theory known as “new massive gravity” (NMG) [3]. In this construction, the mass
scale arises as a consequence of fourth-order terms in the action (curvature-squared
terms in the 3D case). A naive extension of the construction from 3D to 4D fails due
to the ghosts implicit in higher-derivative theories for D > 3. Nevertheless, we have
shown that there is an extension of the NMG construction to D > 3, at least at the
linearized level, if the graviton field is in an “exotic” Lorentz representation, introduced
by Curtright [12] in the context of a “generalized Fierz-Pauli” model. In that context
the dual graviton field is not a gauge field but we have shown that it may be used to
construct an equivalent 4th-order gauge theory for a spin-2 gauge potential in the same
“exotic” Lorentz representation.
Although this construction can be carried out for any spacetime dimension D ≥ 4,
we have focused here on the 4D case. In that case, we have found an explicit fourth-
order action for a dual gauge potential that propagates the five independent modes
of a massive graviton. We have called this (linearized) “new massive dual gravity”
(NMDG). The unitarity of this free-field model is guaranteed by the equivalence of the
equations of motion to the FP equations and the fact that all five propagated modes
are related by rotational invariance.
In the infinite mass limit, the propagating modes decouple. At the Lagrangian
level this can be seen from the fact that only a second-order term survives the limit,
and this propagates no modes, as confirmed by an explicit canonical analysis. This is
very similar to what happens for 3D NMG; in that case the second-order term is the
Einstein-Hilbert term, which does not propagate any modes in 3D.
We have also considered the zero-mass limit. This limit can be taken directly in the
field equations, and the resulting equations can be derived from the “pure” 4th-order
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