2014/01/30 Discussion@U Chicago

Derivative interactions
in nonlinear massive gravity
YAMAUCHI, Daisuke
Research Center for the Early Universe (RESCEU),
U. TokyoReferences

Derivative interactions in de Rham-Gabadze-Tolley massive gravity
Rampei Kimura(RESCEU->New York U) and DY,
PRD88, 084025 (2013), arXiv:1308.0523.

Daisuke Yamauchi (RESCEU)

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Motivation

Can we construct healthy massive gravity
even in nonlinear level?

Daisuke Yamauchi (RESCEU)

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“Linear” massive gravity
 Fierz-Pauli massive gravity [Fierz+Pauli (1939)]
Linearized
Einstein-Hilbert term

Only allowed mass term which does
not have the ghost at linear order(1) Lorentz invariant(2) No ghost at linear order(3) Simple “nonlinear extension” contains a ghost at nonlinear level[Boulware+Deser(1971)]
Daisuke Yamauchi (RESCEU)

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Ghost-free nonlinear massive gravity
 Nonlinear extension of Fierz-Pauli massive gravity[de Rham+Gabadadze+Tolley (2011)]

φa is called Stuckelberg field, which restores general covariance.
Daisuke Yamauchi (RESCEU)

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Interactions for linear theories
 Fierz-Pauli mass term Levi-Civita structure ensures that the Lagrangian is linear in h00.
→ h00 becomes a Lagrange multiplier, which kills the ghost!
 Derivative interaction in FP theory[Hinterbichler(2013)] [see also Folkerts+Pritzel+Wintergerst(2011)]→ h00 becomes a Lagrange multiplier.
Daisuke Yamauchi (RESCEU)

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Our work

Is there any consistent “nonlinear extension”
of derivative interactions
in de Rham-Gabadadze-Tolley massive gravity?

Daisuke Yamauchi (RESCEU)

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Guideline for construction of Lagrangian
To construct “nonlinear” derivative interaction in dRGT massivegravity, we demand the following restrictions:(1) Linearization of hμν reproduces Fierz-Pauli massive gravity.
 Lorentz invariance Free of Boulware-Deser ghost at linear order(2) The cutoff energy scale is Λ3.
 All nonlinear terms below Λ3 have to be eliminated.
(3) The resultant theory does not have a BD ghost.
Daisuke Yamauchi (RESCEU)

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Candidates for derivative interactions
There are a number of candidates for nonlinear interactions, such as Define the tensor Hμν as the covariantization of the metricperturbations as
where the four Stuckelberg fields φa transform as scalars.

 Note : We restrict the form of derivative interactions by using
only a Riemann tensor.
Daisuke Yamauchi (RESCEU)

9Decoupling limit (DL) We can safely ignore vector modes in DL and the helocity-0
mode π of the graviton can be extracted by expressing

 DL is convenient to capture high energy behavior below
the Compton wavelength of massive graviton:

Daisuke Yamauchi (RESCEU)

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Energy scale for derivative interactions
The Lagrangian in DL can be schematically written asnπ=1nπ=2nπ=3…
nπ=nnh=1∞
Λ5nh=2
Λ3
Λ3

Λ4…

Λ3…
Λ3

Λ(3n-1)/(n-1)

The energy scale of nπ=1 terms in the DL is potentially dangerous
and these terms have to be eliminated (requirement (2)).
Daisuke Yamauchi (RESCEU)

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HR order
We start with the lowest order terms in a general form;

To satisfy the requirement (1) : Linearization of hμν reproduces
the FP theory, we require d=-1/2. In terms of Levi-Civita symbol,

Daisuke Yamauchi (RESCEU)

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HR order in DL The lowest order term in DL, (nh,nπ)=(1,1), is given by

nπ=1
nπ=2
nπ=3…
nπ=nTotal derivative!

nh=1∞
Λ5Λ4…
Λ(3n-1)/(n-1)
Daisuke Yamauchi (RESCEU)

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HR order in DL The next order term in DL, (nh,nπ)=(1,2), is not total derivative:→Only way to eliminate this term is to add the next order term:

nπ=1nπ=2nπ=3…
nπ=nnh=1∞
Λ5Λ4…

Λ5 term can be eliminated!

…but it contains nπ=3 term which
is not total derivative:Λ(3n-1)/(n-1)
Daisuke Yamauchi (RESCEU)

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HR order in DL Following the same step as nπ=2, the nπ=3 dangerous term canbe eliminated by adding higher-order terms. Then…

Automatically total derivativenπ=1nπ=2nπ=3…
nπ=nnh=1∞
Λ5Λ4…

We can perform the sameprocedure to eliminate nh=1 termin DL by introducing appropriate
counter terms order by order.

Λ(3n-1)/(n-1)
Daisuke Yamauchi (RESCEU)

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Total HR order Lagrangian The total Lagrangian including counter terms is given by This Lagrangian does not have the terms of the energy scales
bellow Λ3, and nonlinear terms contribute at Λ3.
 We have only on nh=1 term in DL, and K tensor ensures nh=1term to be a total derivative in DL.
Daisuke Yamauchi (RESCEU)

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H2R order The starting point of the Lagrangian is given byThis is the only combination that the lowest orderΛ5 term becomes a total derivative.
 With the same procedure of the previous case, thecounter terms can be resumed by using K tensor again,

Note : Linearization of hμν in unitary gauge gives the “pseudolinear” derivative interaction derived by Hinterbichler (2013).
Daisuke Yamauchi (RESCEU)

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H3R or higher order
We do not have total derivative combination due to the numberof indices of antisymmetric tensor in 4 dim.

There is no higher order Lagrangian satisfying the restrictions.

 The derivative interaction for massive gravity we found is

Daisuke Yamauchi (RESCEU)

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BD ghost???
So far we could successfully eliminate all energy scale below Λ3,
and the derivative interaction in DL contribute at Λ3.
But we still need to check the requirement (3) : BD ghost.
 Λ3 theory in DL (nh=2, nπ=arbitrary)

EoM is 2nd order differential equation
These terms yield 4th orderdifferential equations for h and π…
There appear extra degrees of freedom, which leads to a ghost…
The mass scale of the derivative interactions should be M<Mpl.
Daisuke Yamauchi (RESCEU)

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Summary
We found the nonlinear derivative interaction in dRGT massive gravity.
 The energy scales below Λ3 can be eliminated by adding counter terms.

 The Lagrangian can be resumed by using K tensor.
 Nonlinear terms contribute at Λ3.
 There appear fourth order differential equations of the scalar and tensor
modes in DL, which leads to a ghost....
[see also de Rham+Matas+Tolley (2013)]

Daisuke Yamauchi (RESCEU)

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Other candidates (in progress…)
We found other Λ3 derivative interactions without the Riemanntensor. There exist other derivative interactions using the covariant
derivative of K tensor.

But … so far we cannot kill higher derivative terms in EoM
even if we combine all possible derivative interaction terms.

Daisuke Yamauchi (RESCEU)

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