**Unformatted text preview: **^
^ [email protected] Þn . We succeeded in eliminating
and the rest scalars @A
the scalar self-interactions, but since these always came from
^
^
combinations (A þ @ ) the terms @[email protected] Þn are automati A X ðnÞ , where the X ðnÞ are the functions
cally of the form @
of @ @ described in the Appendix, which are identically
ðn Þ
conserved @ X ¼ 0. Thus, once the scalar self-interactions
^
^
are eliminated, the @[email protected] Þn terms are all total derivatives
and are also eliminated.
Now the lowest interaction scale will be due to the terms in
Eq. (8.3),
$ ^
^
[email protected] Þn
;
nþ1 2nþ2
MP m $ ^
^
[email protected] [email protected] Þn
;
nþ2 2nþ4
MP m (8.8) which are suppressed by the scale Ã3 ¼ ðMP m2 Þ1=3 , so the
cutoff has be raised to Ã3 , carried by the terms (8.8).
This theory can, in fact, be resummed in an interesting
way, using an action involving square roots (de Rham,
Gabadadze, and Tolley, 2010; Hassan and Rosen, 2011b).
The decoupling limit is now
m ! 0; MP ! 1; Ã3 fixed; (8.9) and the only terms which survive are those in Eq. (8.3). To
¨
ﬁnd these terms we must now go back to the full Stuckelberg
replacement (6.31), and we must also expand the inverse
metric and determinant in the potential of Eq. (5.9) in powers
of h. The [email protected] Þn terms, up to quintic order in the decoupling
limit, and up to total derivatives are (de Rham and
Gabadadze, 2010a)
Z
1 ^
ð1 Þ ^
4 1^
; ^
h À h
À4X ðÞ
S ¼ d x h E
2
2
4 ð 6 c 3 À 1 Þ ð2 Þ ^
16ð8d5 þ c3 Þ ð3Þ ^
þ
X ðÞ þ
X ðÞ
Ã3
Ã6
3
3
1^
(8.10)
h T :
þ
MP
ð nÞ
Here the X are the identically conserved combinations of
^
@ @ described in the Appendix. The [email protected] [email protected] Þ terms are
found to cubic order by de Rham and Gabadadze (2010b).
The terms with A’s can in any case be consistently set to zero
at the classical level, since they never appear linearly in the
Lagrangian, so we focus only on the terms involving h and .
de Rham, Gabadadze, and Tolley (2010) used a nice trick to
show that the decoupling limit Lagrangian (8.10) is exact to
all orders in the ﬁelds, that is, there are no further terms
[email protected] Þn for n ! 4. Properties of this Lagrangian, including
its cosmological solutions, degravitation effects, and
phenomenology are studied by de Rham et al. (2010).
Spherical solutions are studied by Chkareuli and
Pirtskhalava (2011). The cosmology of a covariantized version was studied by de Rham and Heisenberg (2011). Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 In terms of the canonically normalized ﬁelds (7.3), the
gauge symmetries (6.34) of the full theory are
2 ^ ^
^
^
^
@ A ;
A ¼ @ Ã À m þ
MP (8.11) 2
^
^
^
h ¼ @ þ @ þ
L ^h ;
MP (8.12) ^
¼ ÀmÃ; (8.13) ^
^
where we rescaled Ã ¼ ðmMP =2ÞÃ and ¼ ðMP =2Þ . In
the decoupling limit (8.9), this gauge symmetry reduces to its
linear form
^
^
A ¼ @ Ã; (8.14) ^
^
h ¼ @ þ @ ; (8.15) ¼ 0: (8.16) The Lagrangian (8.10) should be inv...

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