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the Vainshtein radius. Below the Vainshtein radius (high
energies), a new degree of freedom, the ghost (analogous to
the physical Higgs in the standard model), kicks in. Much
below the Vainshtein radius, everything is again linear and
weakly coupled, with the difference that there are now 2 active
degrees of freedom, so one can think of this as a classical UV
completion of the effective theory.
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 695 Of course, this ghostly mechanism for restoring continuity
with GR relies on an instability, which would become
apparent were we to investigate small ﬂuctuations beyond
the gross-scale features described here. Furthermore, as we
see in the next section, the ghost issue is moot, since the
classical mechanism described in this section occurs outside
the regime of validity of the quantum effective theory and is
swamped by unknown quantum corrections.
D. Quantum corrections and the effective theory Quantum mechanically, massive gravity is an effective
ﬁeld theory, since there are nonrenormalizable operators suppressed by the mass scale Ã5 . The amplitude for !
scattering at energy E, coming from the cubic coupling in
Eq. (7.10), is similar to A $ ðE=Ã5 Þ10 . This amplitude
should correspond to the scattering of longitudinal gravitons.
The wave function of the longitudinal graviton (2.20) for a
large boost is proportional to mÀ2 , while the largest term at
high momentum in the graviton propagator (2.44) is proportional to mÀ4 , so naive power counting suggests that the
amplitude at energies much larger than m is similar to A $
2
E14 =MP m12 . However, as recognized by Arkani-Hamed,
Georgi, and Schwartz (2003) and calculated explicitly by
Aubert (2004), there is a cancellation in the diagrams so
¨
that the result agrees with the result of the Stuckelberg
description. We encounter these kinds of cancellations again
¨
in loops, and part of the usefulness of the Stuckelberg description is that they are made manifest.
The amplitude becomes of the order of 1 and hence
strongly coupled when E $ Ã5 . Thus Ã5 is the maximal
cutoff of the theory. We expect to generate all operators
compatible with the symmetries, suppressed by appropriate
powers of the cutoff. In the unitary gauge, there are no
symmetries, so we generate all operators of the form
cp;q @q hp : (7.19) We determine the scales in the coefﬁcient cp;q .
¨
After Stuckelberging, the decoupling limit theory contains
^
only the scalar and the single coupling scale Ã5 . In
^
^
addition, there is the Galileon symmetry ! þ c þ
c x . Quantum mechanically, we expect to generate in the
quantum effective action all possible operators with this
symmetry, suppressed by the appropriate power of the cutoff
^
Ã5 . The Galileon symmetry forces each to carry at least
two derivatives,13 so the general term we can have is
$ 13 ^
@q ð@2 Þp
Ã3pþqÀ4
5 : (7.20) Actually, there are a ﬁnite number of terms which have fewer
than two deriva...

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