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**Unformatted text preview: **only 5 degrees of
freedom.
Following Creminelli et al. (2005), let us consider the
stability of the classical solutions to Eq. (7.10) around a
massive point source. We have a classical background ÈðrÞ,
^
which is a solution of the equation of motion, and we
expand the Lagrangian of Eq. (7.10) to quadratic order in the
^
ﬂuctuation ’ À È. The result is schematically
L’ $ Àð@’Þ2 þ ð@2 ÈÞ 2 2
ð@ ’Þ :
Ã5
5 Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (7.15) There is a four-derivative contribution to the ’ kinetic term,
signaling that this theory propagates 2 linear degrees of
freedom. As shown in Sec. 2 of Creminelli et al. (2005),
one is stable and massless, and the other is a ghost with a
mass of the order of the scale appearing in front of the higher
derivative terms. So in this case the ghost has an r-dependent
mass
m 2 ð rÞ $
ghost Ã5
5
:
@2 ÈðrÞ (7.16) This shows that around a ﬂat background, or far from the
source, the ghost mass goes to inﬁnity and the ghost freezes,
explaining why it was not seen in the linear theory. It is only
around nontrivial backgrounds that it becomes active. Notice,
however, that the backgrounds around which the ghost becomes active are perfectly nice, asymptotically ﬂat conﬁgurations sourced by compact objects such as the Sun, and not
disconnected in any way in ﬁeld space (this is in contrast to
the ghost in DGP, which occurs around only asymptotically
de Sitter solutions).
We are working in an effective ﬁeld theory with a UV
cutoff Ã5 ; therefore we should not worry about instabilities
until the mass of the ghost drops below Ã5 . This happens at
the distance rghost where @2 Èc $ Ã3 . For a source of mass M,
5
at distances r ) rV the background ﬁeld is similar to ÈðrÞ $
ðM=MP Þð1=rÞ, so
1=3
1=5
M
1
M
1
) rV $
:
(7.17)
rghost $
MP
Ã5
MP
Ã5
rghost is parametrically larger than the Vainshtein radius rV .
As we see in Sec. VII.D, the distance rghost is the same
distance at which quantum effects become important.
Whatever UV completion takes over should cure the ghost
instabilities that become present at this scale, so we will be
able to consistently ignore the ghost. We see already that we
cannot trust the classical solution even in regions parametrically farther than the Vainshtein radius. The best we can do is
make predictions outside rghost , and we have more to say
about this later.
C. Resolution of the vDVZ discontinuity and the Vainshtein
mechanism We are now in a position to see the mechanism by which
nonlinearities can resolve the vDVZ discontinuity. This is
known as the Vainshtein mechanism. It turns out to involve
the ghost in a critical role.
Far outside the Vainshtein radius, where the linear term of
Eq. (7.10) dominates, the ﬁeld has the usual Coulombic 1=r
form. But inside the Vainshtein radius, where the cubic term
dominates, it is easy to see by power counting that the ﬁeld
gets an r3=2 proﬁle,
8
M
>M 1;
r ) rV ;
< Pr
^
$ ...

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