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**Unformatted text preview: **ariant under the decoupling limit gauge symmetries (8.16). Indeed, the identity
ðn Þ
@ X ¼ 0 ensures that it is. The scalar is gauge invariant
in the decoupling limit, but the fact that it always comes with
two derivatives means that the global Galileon symmetry
(7.8) is still present, as is the shift symmetry on A .
Note that for the speciﬁc choices c3 ¼ 1=6 and d5 ¼
À1=48, all the interaction terms disappear. This could mean
that the theory becomes strongly coupled at some scale larger
than Ã3 , or there could be no lowest scale, since there are
scales arbitrarily close to but above Ã3 . In the latter case, the
theory would have no nonlinear behavior, and so no mechanism to recover continuity with GR, and it would therefore be
ruled out observationally.
B. The appearance of Galileons and the absence of ghosts We partially diagonalize the interaction terms in Eq. (8.10)
by using the properties (A.18). First we perform the conformal transformation needed to diagonalize the linear terms,
^
^
^
h ! h þ , after which the Lagrangian takes the
form
Z
1^
^
S ¼ d4 x h E ; h 2
1^
4ð6c3 À 1Þ ^ ð2Þ 16ð8d5 þ c3 Þ ^ ð3Þ
X þ
X
À h
2
Ã3
Ã6
3
3
þ 1^
6 ð 6 c3 À 1 Þ ^ 2 ^
^
ð@Þ h
h T À 3ð@Þ2 þ
MP
Ã3
3 þ 16ð8d5 þ c3 Þ ^ 2 ^ 2
1^
^
ð@Þ ð½Å À ½Å2 Þ þ
T:
MP
Ã6
3
(8.17) ^
^
Here the brackets are traces of Å @ @ and its powers
(the notation is explained at the end of the Introduction).
The cubic h couplings can be eliminated with a ﬁeld
redeﬁnition
2 ð 6 c3 À 1 Þ
^^
^
^
@ @ ;
h ! h þ
Ã3
3 Kurt Hinterbichler: Theoretical aspects of massive gravity after which the Lagrangian reads
S¼ Z 1^
8ð8d5 þ c3 Þ ^ ^ ð3Þ
^
d4 x h E ; h À
h X
2
Ã6
3 þ 1^
6ð6c3 À 1Þ ^ 2 ^
^
ð@Þ h
h T À 3ð@Þ2 þ
MP
Ã3
3 À4
À ð6c3 À 1Þ2 À 4ð8d5 þ c3 Þ ^ 2 ^ 2
^
ð@Þ ð½Å À ½Å2 Þ
Ã6
3 40ð6c3 À 1Þð8d5 þ c3 Þ ^ 2 ^ 3
^
^
ð@Þ ð½Å À 3½Å2 ½Å
Ã9
3 1^
2 ð 6 c3 À 1 Þ
^
^^
þ 2½Å3 Þ þ
@ @ T :
T þ
MP
Ã3 MP
3
(8.18)
There is no local ﬁeld redeﬁnition that can eliminate the
h quartic mixing (there is a nonlocal redeﬁnition that
can do it), so this is as unmixed as the Lagrangian can get
while staying local.
The scalar self-interactions in Eq. (8.18) are given by the
following four Lagrangians:
L2 ¼ À1ð@Þ2 ;
2 L3 ¼ À1ð@Þ2 ½Å;
2 L4 ¼ À1ð@Þ2 ð½Å2 À ½Å2 Þ;
2
L5 ¼ À1ð@Þ2 ð½Å3
2 (8.19) À 3½Å½Å þ 2½Å Þ:
2 3 These are known as the Galileon terms (Nicolis, Rattazzi, and
Trincherini, 2008) [see also Sec. II of Hinterbichler, Trodden,
and Wesley (2010) for a summary of the Galileons]. They
share two special properties: their equations of motion are
purely second order (despite the appearance of higher derivative terms in the Lagrangians), and they are invariant up to a
total derivative under the Galilean symmetry (7.8), ðxÞ !
ðxÞ þ c þ b x...

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