Gravitational- and Self- Coupling of Partially Massless Spin 2

0.2 MB·English

by  S. Deser

CALT 68-2884
BRX-TH-661
Gravitational- and Self- Coupling of Partially
3
1 Massless Spin 2
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2

n
a S. Deser♯, E. Joung♭ and A. Waldron♮
J

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1 ♯Lauritsen Lab, Caltech, Pasadena CA 91125 and Physics Department, Brandeis

University, Waltham, MA 02454, USA
]
h [email protected]
t
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p
♭ Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126 Pisa, Italy
e
h [email protected]
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1 ♮Department of Mathematics, University of California, Davis, CA 95616, USA
v
1 [email protected]
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1
4
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Abstract
0
3
1 We show that higher spin systems specific to cosmological spaces are
:
v subject to the same problems as models with Poincar´e limits. In
i
X particular, we analyse partially massless (PM) spin 2 and find that
r both its gravitational coupling and nonlinear extensions suffer from
a
the usual [background- and self-coupling] difficulties: Consistent free
field propagation does not extend beyond background Einstein ge-
ometries. Then, using conformal Weyl gravity (CG), which consists
of relative ghost PM and graviton excitations, we find that avoiding
graviton-ghosts restrictsCG-generatedPMself-couplingstotheusual,
safe, Noether current cubic ones.
PACS: 04.62.+v, 04.50.-h, 04.50.Kd, 04.62.+v, 02.40.-k
1 Introduction
Theconsistency difficulties ofmassless andmassive higher spinfields ind = 4
are by now well-explored, both regarding their coupling to gravity and other
fields as well as possible self-interactions. Our aim here is to investigate
these problems for partially massless (PM) theories [1, 2], which have the
novel feature that their (anti) de Sitter ((A)dS) higher spin representations
havenodirectPoincar´ecounterparts. ForthisweemployWeyl—conformal—
gravity (CG) as a tool. Even though CG is physically unacceptable (being
fourth derivative order, its physical excitations are relatively ghost-like) it
can be safely used when one of its two, graviton and PM [3], components can
be fixed, while studying the other [4].
We will begin by reviewing PM and then show that it precisely char-
acterizes CG solutions that are not conformally Einstein spaces. We then
explain, using recent mathematical tools, how CG can be safely exploited for
our consistency analyses of PM. The first question:
whatare the mostgeneralgeometricalfixed backgroundsin whichPM
consistently propagates?
can then be answered–they are essentially restricted to Einstein spaces. The
second consistency question:
can one define a self-interacting version of the free field, even in
Einstein vacuum?
willthenbeaddressed, yieldingaminortriumphaswell: onlytheusualcubic,
abelianNoether current-field coupling isgenerated viaCG.Weconclude with
speculations regarding PM’s possible cosmological and formal uses.
2 Review of PM and its CG embedding
The PM tensor field ϕ dynamics are defined in any Einstein background
µν
by the action
= √ g 1 ϕ 2 νϕ 2 + µϕ νϕ 1 ϕ 2 (1)
− − 2 ∇ρ µν − ∇ µν ∇ ∇ µν − 2 ∇µ
R h +ϕ(cid:0)µνW (cid:1)ϕρσ(cid:0)+ 2Λ ϕ(cid:1) 2 1ϕ2 , ϕ :(cid:0)= ϕρ,(cid:1) (2)
ρµνσ 3 µν − 4 ρ
i
(cid:2)(cid:0) (cid:1) (cid:3)
2
and field equations
∆ϕ 2 ρϕ +g ρ σϕ + ϕ g ∆ϕ
µν (µ ν)ρ µν ρσ µ ν µν
− ∇ ∇ ∇ ∇ ∇ ∇ −
2W ϕρσ 4 Λ(ϕ 1 g ϕ) = 0,
− ρµνσ − 3 µν − 4 µν
where W is the Weyl tensor. This system is invariant under a double
µνρσ
derivative gauge transformation
δϕ = ∂ + Λ g α(x), (3)
µν ∇µ ν 3 µν
(cid:0) (cid:1)
which is the tuned sum of a metric fluctuation diffeomorphism (with param-
eter ∂ α(x)) and a conformal transformation. This system is a hybridization
µ
ofstrictly massless andnormal massive, Fierz-Pauli, spin 2. Indeed, there are
three varieties of spin 2 excitations in dS: massive, massless and PM [1, 2].
IndS,PMpropagateslightlike, positive energy(inside themaximally accessi-
ble intrinsic horizon), helicity 2, 1 excitations in a unitary representation
± ±
of the isometry group [5, 7, 6, 8]. This degree of freedom (DoF) count relies
on the gauge invariance (3) and the divergence constraint µϕ = ϕ
µν ν
∇ ∇
implied by integrability of (1).
Interactions of PM in four dimensions are particularly interesting because
it is rigidly SO(4,2) conformally invariant [9], just like its vector Maxwell
counterpart. In fact, PM can be coupled to charged matter fields [10] (see
also [11]). [Forming non-abelian multiplets is still an openproblem.] Instead,
we will be concerned with its self interactions, whose cubic vertices were first
given in [12] using a Stu¨ckelberg approach.1
Since Weyl transformations underlie PM’s invariances (see (3)), CG is
a natural tool for studying its interactions. While CG always has six exci-
tations, the detailed spectra are background-dependent. About flat space,
it has two massless tensors and a photon with the same signature as one of
them [15], while in constant curvature backgrounds there is still a (cosmolog-
ical) graviton, but now the (tensor+photon) combination becomes the PM
mode with helicities ( 2, 1). In each case, the two sets of modes are rela-
± ±
tively ghost-like. The relative sign between PM’s helicities depends on that
of Λ: In AdS, one can truncate the solution space to just the unitary, mass-
less graviton [3, 16, 17] (for related analysis of higher derivative theories
1A generalcalculus of higher derivative PM cubic verticeswas developedin [13]. Also,
it has recently been suggested that a PM limit of putative massive gravity theories could
be a candidate for an interacting PM theory [14].
3
see [18, 19]). The dS story is the more interesting one because we can trun-
cate, leaving either mode unitary; keeping the unitary PM mode, is the
relevant case here.
The CG action is
S[g] = 1 √ g WµνρσW = 1 √ g Rµν R 1 R2 , (4)
8 − µνρσ 4 − µν − 3
Z Z
(cid:0) (cid:1)
and its field equation is the vanishing of the Bach tensor,
1 1
B := ∆P + ρ P +W Pρσ, P := (R g R).
µν µν (µ ν)ρ ρµνσ µν µν µν
− ∇ ∇ 2 − 6
P
The Schouten tensor measures the difference between Riemann and Weyl
µν
tensors, R W = g P g P +g P g P , and is a main-
µνρσ µνρσ µρ νσ νρ µσ νσ µρ µσ νρ
− − −
stay of conformal models in all dimensions: its variation is a pure (double)
gradient,
δg = 2αg δP = ∂ α.
µν µν µν µ ν
⇒ −∇
The Bach tensor B is, of course, invariant under this rescaling. For our
µν
purposes, it is more convenient to work with the cosmological Schouten ten-
sor,
ϕ := P + Λ g , (5)
µν − µν 6 µν
in terms of which B reads
µν
B (g,ϕ) = ∆ϕ 2 ρϕ +g ρ σϕ + ϕ g ∆ϕ
µν µν (µ ν)ρ µν ρσ µ ν µν
− ∇ ∇ ∇ ∇ ∇ ∇ −
2W ϕρσ 4 Λ(ϕ 1 g ϕ) + O ϕ2 . (6)
− ρµνσ − 3 µν − 4 µν
(cid:0) (cid:1)
Consider now configurations such that the metric is close to an Einstein one
with cosmological constant Λ (it is important to note that the set of Bach
flat, but non-Einstein metrics is non-empty, see [20]). Then, by (5), ϕ is
µν
a small excitation and its field equation (6) is precisely the PM one in this
background, the Schouten tensor’s Weyl transformation implying the PM
gaugeinvariance (3). We have now recovered CG’s (linearized) PM subsector
by holding the metric constant (or in other words, setting the metric to a
non-dynamical background field). This key fact motivates our use of CG
as a probe of PM for two basic higher spin questions: How general are the
geometries in which it can propagate consistently? Does CG provide a useful
starting-point for studying possible self-interactions of PM?
4
We next answer the first question: we use CG to generate a list of in-
creasingly general metrics, from dS to Einstein to Bach, and show that there
is indeed a natural barrier–one that is much closer to Einstein than, as one
might reasonably conjecture, to Bach.
3 PM in a background
It has been established that there exist Weyl invariant field equations en-
joying a double derivative gauge invariance in Bach-flat backgrounds [22].
This result suggests that Bach-flat is the most general background support-
ing consistent (linear) PM propagation. In detail, the operator from scalars
to trace-free symmetric tensors,
P := ∂ +P ,
µν {µ ν} {µν}
∇
permits a factorization of the Bach tensor as
B = MρσP , (7)
µν µν ρσ
Mρσ := δρ δσ ∆ δρ σ 1 δρ σ Wρ σ.
µν {µ ν} − {µ∇ ∇ν} − 3 {µ∇ν}∇ − µν
We observe that Mρσ gives the non-linear answer to the question posed in
µν
the Introduction: characterizing Bach-flat metrics that are not conformally
Einstein (the latter are characterized in [23] and [24]). We see that those
require the range of P to intersect the kernel of M ; the operator M is
µν µν µν
also conformally invariant and maps trace-free symmetric tensors to trace-
free symmetric tensors. Physically, it implies that the field equation
Mρσϕ = ∆ϕ σ ϕ 1 σϕ Wρ σϕ = 0, (8)
µν ρσ µν −∇ ∇{µ ν}σ − 3 ∇{µ∇ ν}σ − µν ρσ
for a trace-free symmetric tensor ϕ =: ϕ = ϕ 1g ϕ, enjoys the
e e e µν {µν}e µν − 4 µνe
double derivative gauge invariance (and associated double derivative Bianchi
identity)
e
δϕ = P α = +P α,
µν µν {µ ν} µν
∇ ∇
inBach-flat backgrounds. This wasth(cid:0)emotivationfo(cid:1)rouroriginalconjecture
e
that PM fields could peropagate in them. We now proceed to disprove it and
give necessary consistency conditions for PM-compatible backgrounds.
The Bach tensor, since it arises from a metric variational principle, is
necessarily divergence-free, µMρσP = 0. However, it is neither true
∇ µν ρσ
5
that µMρσ = 0, nor even that µMρσ = O( ) (rather this operator is
∇ µν ∇ µν ∇
cubic in derivatives). But consistent PM propagation relies on a divergence
constraint2; foraPMfieldequation(derived fromanaction) thisrequirement
is precisely expressed by the condition µMρσ = O( ).
∇ µν ∇
The failure of the field equation (8) to imply an appropriate divergence
constraint does not yet rule out PM fields interacting with backgrounds more
general than Einstein spaces, because we may still enlarge the space of field
equation and gauge operators, Mρσ and P respectively, by relaxing their
µν µν
trace-free and conformal invariance properties. To test this, we make the
following generalization
M′ρσ = Gρσ δρ δσ g gρσ P+α δρ Pσ +α g Pρσ +P gρσ ,
µν µν − (µ ν) − µν 1 (µ ν) 2 µν µν
Pµ′ν = ∇µ∂ν +(cid:0) 12 Pgµν +βPµν ,(cid:1) e (cid:0) e e (cid:1)
where the cosmological Einsteien operator
Gρσ := δρ δσ g gρσ ∆ P 2 ρδσ + gρσ +g ρ σ
µν (µ ν) − µν − − ∇(µ∇ ν) ∇(µ∇ν) µν∇ ∇
(cid:0) 2Wρ σ 8P(cid:1)ρ(cid:0)δσ 3(cid:1)g Pgρσ, (9)
− µν − {µ ν} − 2 µν
is identically conserved, e
µGρσ = 0,
∇ µν
in Einstein backgrounds. The equation of motion of cosmological Einstein
gravity linearized about an Einstein metric is Gρσ ϕ = 0.
µν ρσ
The above ansatz is the most general one obeying the following require-
ments:
1. The operators M′ρσ and P′ are second order in or derivatives on
µν µν ∇
the metric g .
µν
2. The operator M′ρσ is self-adjoint, to ensure the existence of an action
µν
principle.
3. The divergence µM′ρσ is an operator no more than linear in so
∇ µν ∇
that solutions of M′ρσ ϕ = 0 obey a first order constraint.
µν µν
2The DoF count for PM starts with ten off-shell fields ϕ , minus four DoF thanks
µν
to the divergence constraint µϕ = ϕ, minus two further DoF by the scalar gauge
µν ν
∇ ∇
invariance, yielding a total of four on-shell excitations.
6
4. The operator product M′ρσP′ vanishes when g is an Einstein met-
µν ρσ µν
ric; this fixes their leading terms to be operators corresponding to the
linear PM equation of motion (6) and its double derivative gauge in-
variance (3). The remaining freedom in the ansatz therefore depends
P
only on the trace-free Schouten tensor , since that quantity vanishes
µν
for Einstein metrics.
e
It remains to compute the product M′ρσ P′ . The result can be ar-
µν ρσ
ranged as an expansion in the gradient operator . By construction, terms
∇
of order 4 and 3 necessarily vanish. Prefactors of the terms order 2
∇ P ∇ ∇
only involve which we are now assuming to be non-vanishing, since we
µν
wish to investigate metrics that are not Einstein: we must choose the con-
stants (α ,αe,β) accordingly and find
1 2
α = 4+2β, α = β.
1 2
−
The analysis of terms order and lower is more complicated. First we
∇
consider the trace gµνM′ρσP′ at order and find 3β( P) ρ. There are
µν ρσ ∇ ∇ρ ∇
two possibilities, either β = 0 or the background metric has constant scalar
curvature. Since the latter would rule out the PM conjecture in question,
we choose β = 0. We then find gµνM′ρσP′ = 3(∆P), which requires the
µν ρσ −
scalar curvature to be harmonic, and hence also rules out the conjecture.
Having excluded Bach-flat backgrounds, we may still investigate whether
some condition stronger than Bach-flat, but still less stringent than Ein-
stein, could yield consistent propagation. The terms remaining at order
∇
in M′ρσP′ are
µν ρσ
βg ( P) ρ (β 2)( P) +2(β 1)( P ) ρ 2β( P ) ρ;
µν ρ (µ ν) ρ µν (µ ν)ρ
∇ ∇ − − ∇ ∇ − ∇ ∇ − ∇ ∇
clearly no choice of β removes all of them. Instead, we can restrict the
P
background, one option being to Ricci-symmetric spaces, defined by
ρ µν
∇
= 0. This condition is weaker than Einstein, but need not imply Bach-flat.
However, even then we must cancel all terms in M′ρσP′ of order 0. In
µν ρσ ∇
general backgrounds these are
βB +2β2Pρ P 1(β 1)(β +3)PP 1(β 2) ∂ P
− µν (µ ν)ρ − 2 − µν − 2 − ∇µ ν
+g 1(β 2)∆P β(β +1)P Pρσ + 1(β +2)(3β 2)P2 .
µν 2 − − ρσ 8 −
EvenforaR(cid:2)icci-symmetricspace(wherethederivativetermsdrop),(cid:3)nochoice
of β removes all remaining terms quadratic in the Schouten tensor and its
7
trace. [We see no strong physical motivation to single out backgrounds with
covariantly constant Einstein tensor subject to a further quadratic curvature
constraint.] This last detour reassures us that no interesting, at best slightly
more general than Einstein, backgrounds are allowed.
4 PM self-interaction?
We emphasize at the outset that the aim of this section is to study putative
self-interacting extensions of PM solely within the context of the CG frame-
work. That is, our results—which will face the usual stringent limitations
on such extensions—strictly apply only to this framework, although they are
suggestive, and the allowed nonlinearities are quite efficiently generated. We
will need a version of the CG action that is more useful for our purposes,
in which the PM field is clearly isolated. This is accomplished by CG’s
“Ostrogradsky” second order formulation [21],
S[g,ϕ] = √ g Λ (R 2Λ)+ϕµν Γ +ϕµν ϕ ϕ2 ,
− − 6 − µν µν −
Z
h i
where Γ := G + Λg is the cosmological Einstein tensor. Upon com-
µν µν µν
pleting the square, we see that the auxiliary field becomes the cosmological
Schouten tensor (5). To analyze the spectrum of the theory about an Ein-
stein background g¯ with cosmological constant Λ, we linearize in metric
µν
perturbations h = g g¯ . Keeping terms quadratic in fluctuations and
µν µν µν
−
making the field redefinition
h h + 6 ϕ , (10)
µν → µν Λ µν
yields the action (the metrics appearing in G and F are set to g¯ )
µν
S(2)[h,ϕ] = 1 √ g¯ Λ hµν Gρσh 6 ϕµν Gρσ 2 ΛFρσ ϕ . (11)
−4 − 6 µν ρσ − Λ µν − 3 µν ρσ
Z h i
(cid:0) (cid:1)
Here Gρσ h /2 is the linearized cosmological Einstein tensor defined in (9)
− µν ρσ
and all indices are moved by g¯ . The Pauli–Fierz (PF) mass operator is
µν
defined as Fρσ := δρδσ g gρσ, so the PM field equation is
µν µ ν − µν
Gρσ 2 ΛFρσ ϕ = 0.
µν − 3 µν ρσ
(cid:0) (cid:1)
8
Thus, the first term of (11) is linearized Einstein–Hilbert, while the terms
with round brackets (the sum of the linearized gravity kinetic term and a
Pauli–Fierz mass term tuned to the PM value m2 = 2Λ/3) give the PM
theory, all in an Einstein background. Hence the model describes the “differ-
ence” of massless and PM excitations. Moreover, integrating out (at linear
level) the field ϕ appearing before the field redefinition (10), gives the
µν
fourth order equation
Bρσh = 0, where Bρσ := GαβF−1γδ Gρσ 2 ΛGρσ ,
µν ρσ µν µν αβ γδ − 3 µν
for the original metric fluctuations. Indeed, Bρσh is the Bach tensor lin-
µν ρσ
earized about an Einstein background.
The relative sign of the two parts of the linearized action (11) reflects
the unavoidable relative ghost structure. In particular, states with ϕ = 0
µν
constitute a unitary, massless spin s = 2 spectrum. When the cosmological
constant is positive (dS), states with h = 0 correspond to a unitary PM
µν
spectrum. We now proceed to study the latter truncation; a key step is to
understand the model’s gauge structure. At linear level, the graviton h
µν
enjoys a linearized diffeomorphism symmetry3 δh = ξ + ξ while
µν µ ν ν µ
∇ ∇
the PM field ϕ transforms according to the double derivative scalar varia-
µν
tion (3); at linear level each field is inert under the other’s transformations.
In fact, the PM gauge symmetry is inherited from the Weyl symmetry of CG.
The full non-linear action (4) is invariant under both gauge transformations,
δg = ξ + ξ +2αg ,
µν µ ν ν µ µν
∇ ∇
δϕ = ϕ + ∂ + Λ g α. (12)
µν Lξ µν ∇µ ν 3 µν
The metric transformation is now a su(cid:0)m of diffeomor(cid:1)phism and Weyl trans-
formations as is the ϕ transformation: is the Lie derivative along the
µν ξ
L
vector field ξ and the Weyl term follows from the transformation of the
Schouten tensor (2).
3As an aside, we observe that the derivation of the linear PM model from Weyl invari-
antCGtheorygivesanovelproofofthe SO(4,2)conformalinvarianceofPMexcitations.
(In fact, conformal invariance was the original rationale behind the PM model [1], and is
enjoyedbyallmaximaldepth,four-dimensionalPMtheoriesofgenericspin[9].) Indetail,
whenever a field is coupled to the metric, maintaining Weyl invariance, then setting the
metric to a background yields an action that enjoys any conformal isometries as symme-
tries. Thusthe non-linearmodelgeneratedby settingthe metricin(4)toabackgroundis
guaranteedtoenjoythissymmetry;sinceitholdsorderbyorderinϕ,itisalsoasymmetry
of linearized PM.
9
Without incurring the ghost problem of CG, we may search for some
combination of fields that, when held to an appropriate background, yields a
consistent truncation to a self-interacting PM model.4 We must now find the
proper combination of fields to set to a background that yields the desired
decoupling. At linear level, the answer to this requirement is given by the
field redefinition (10). There, the choice for the metric fluctuations h = 0 is
µν
respected by PM gauge transformations. This substitution in the linearized
action (11) yields the free PM action in an Einstein background. Therefore
we begin by positing a candidate for a non-linear version of the field redefini-
tion (10) (that mixes g and ϕ ) such that a consistent PM theory results
µν µν
from holding the redefined metric to a suitable fixed value:
g g + 6 ϕ
µν → µν Λ µν (13)
( ϕµν ϕµν .
→
[We could have allowed for further redefinitions of both fields, by adding (to
each) initially arbitrary functions starting at second order, so as to preserve
the linear choice (10), but in fact this would only affect quartic corrections,
and we will, for good reason, stop at cubic order.] With this field redefini-
tion, the CG action (4) reduces to that of a “matter” field ϕ coupled to a
µν
(dynamical) metric:
S[g,ϕ] = √ g Λ (R 2Λ)+ 6 L (ϕ, ϕ) ,
− − 6 − Λ PM ∇
Z
(cid:2) (cid:3)
where L is the candidate PM Lagrangian. Its ϕ dependence is highly
PM µν
non-linear, with self-interactions coming from re-expressing all the original
metric dependence of the action (4) in terms of the shifted combination g +
µν
6 ϕ . After making this expansion, we set g to any Einstein metric with
Λ µν µν
cosmological constant Λ. This leaves us with the PM candidate
S [ϕ] = 6 √ gL (ϕ, ϕ),
PM Λ − PM ∇
Z
4Indeed, the converseversionofthis procedure canbe appliedto produce cosmological
gravityfromCGforthefull,non-lineartheory: Examiningthegaugetransformations(12),
we see that the PM background ϕ = 0 is preserved by diffeomorphisms but not Weyl
µν
transformations. Hence, setting ϕ = 0 yields a diffeomorphism invariant theory; per-
µν
forming this substitution in the action (4) gives cosmologicalEinstein gravity.
10