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**Unformatted text preview: **e Vainshtein mechanism in the Ã3 theory In the Ã5 theory, the key to the resolution of the vDVZ
discontinuity and recovery of GR was the activation of the
Boulware-Deser ghost, which canceled the force due to the
longitudinal mode. In the Ã3 theory, there is no ghost (at least
in the decoupling limit), so there must be some other method
by which the scalar screens itself to restore continuity with
general relativity. This method uses nonlinearities to enlarge
the kinetic terms of the scalar, rendering its couplings small.
To see how this works, consider the Lagrangian in the
form (8.17). Set d5 ¼ Àc3 =8, c3 ¼ 5=36 to simplify coefﬁcients, and ignore for a second the cubic h coupling, so
that we have only a cubic self-interaction governed by the
Galileon term L3 ,
Z
1
1^
^
^
^
S ¼ d4 x À 3ð@Þ2 À 3 ð@Þ2 h þ T:
(8.27)
M4
Ã
This is the same Lagrangian studied by Nicolis and Rattazzi
(2004) in the DGP context.
Consider the static spherically symmetric solution
^
ðrÞ around a point source of mass M, T $ M3 ðrÞ. The
solution transitions, at the Vainshtein radius rð3Þ
V
ðM=MPl Þ1=3 ð1=Ã3 Þ, between a linear and nonlinear regime.
For r ) rð3Þ the kinetic term in Eq. (8.27) dominates over the
V
cubic term, linearities are unimportant, and we get the usual
1=r Coulomb behavior. For r pﬃﬃﬃ rð3Þ , the cubic term is domi(V
nant, and we get a nonlinear r potential,
8 3 ð3Þ2 r 1=2
< Ã3 rV ð ð3Þ Þ
r ( r ð3 Þ ;
V
rV
^ ð rÞ $
(8.28)
: 3 ð3Þ2 rðV3Þ
r ) r ð3 Þ :
Ã 3 rV ð r Þ
V Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 T ¼ T0 þ T; ^
¼ È þ ’; (8.30) we have after using the identity ð@ ’Þh’ ¼ @ ½@ ’@ ’ À
1
2
2 ð@’Þ on the quadratic parts and integrating by parts
S’ ¼ Z 2
ð@ @ È À hÈÞ@ ’@ ’
Ã3
1
1
À 3 ð@’Þ2 h’ þ ’T:
(8.31)
M4
Ã
d4 x À 3ð@’Þ2 þ Note that expanding the cubic term yields new contributions
to the kinetic terms, with coefﬁcients that depend on the
background. Unlike the Ã5 Lagrangian (7.10), no higher
derivative kinetic terms are generated, so no extra degrees
of freedom are propagated on any background. This is a
property shared by all the Galileon Lagrangians (8.19)
(Endlich et al., 2010).
Around the solution (8.28), the coefﬁcient of the kinetic
term in Eq. (8.31) is Oð1Þ at distances r ) rð3Þ , but goes as
V
ðrð3Þ =rÞ3=2 for distances r ( rð3Þ . Thus the kinetic term is
V
V
enhanced at distances below the Vainshtein radius, which
means that after canonical normalization the couplings of
the ﬂuctuations to the source are reduced. The ﬂuctuations ’
effectively decouple near a large source, so the scalar force
between two small test particles in the presence of a large
source is reduced, and continuity with GR is restored. A more
careful study of the Vainshtein screening in the Ã3 theory,
including numerical solutions of the decoupling limit action,
can be found in Chkareuli and Pirts...

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