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**Unformatted text preview: **ry makes no observable predictions within its range of validity. Kurt Hinterbichler: Theoretical aspects of massive gravity so there is no hope of ﬁnding the leading quantum corrections. Finally, the radius rQ is the same as the radius rghost ,
where the ghost mass drops below the cutoff, so it is consistent to ignore the ghost since it lies beyond the reach of the
quantum effective theory. The various regions are shown in
Fig. 1. Note that in the decoupling limit we are working in,
the Schwarzschild radius (and the radii associated to all scales
larger than Ã5 ) are sent to zero, while the scale r $ 1=m
where Yukawa suppression takes hold is sent to inﬁnity.
VIII. THE Ã3 THEORY We have seen that the theory (5.1) containing only the
linear graviton mass term has some undesirable features,
including a ghost instability and quantum corrections that
become important before classical nonlinearities can restore
continuity with GR. In this section, we consider the higher
order potential terms in Eq. (5.3) and ask whether they can
alleviate these problems. It turns out that there is a special
choice of potential that cures all these problems, at least in the
decoupling limit.
This choice also has the advantage of raising the cutoff.
With only the Fierz-Pauli mass term, the strong coupling
cutoff was set by the cubic scalar self-coupling
^
[email protected] Þ3 =Ã5 . The cutoff Ã5 ¼ ðMP m4 Þ1=5 is very low, and
5
as we see generically any interaction term will have this
cutoff. But by choosing this special tuning of the higher order
interactions, we end up raising the cutoff to the higher scale
Ã3 ¼ ðMP m2 Þ1=3 .
Arkani-Hamed, Georgi, and Schwartz (2003) already recognized that if the scalar self-interactions could be eliminated, the cutoff would be raised to Ã3 . This was studied
more fully by Creminelli et al. (2005), where the cancellation
was worked through and it was (mistakenly) concluded that
ghosts would be unavoidable once the cutoff was raised.
Motivated by constructions of massive gravity with auxiliary
extra dimensions (Gabadadze, 2009; de Rham, 2010; de
Rham and Gabadadze, 2010b), this was revisited by de
Rham and Gabadadze (2010a) and de Rham, Gabadadze,
and Tolley (2010), where the decoupling limit Lagrangian
was calculated explicitly and was seen to be ghost free. The
full theory was shown to be ghost free by Hassan and Rosen
(2011a, 2011c). Looking back at the scales (7.5), the term suppressed by the
smallest scale is the cubic scalar term, which is suppressed by
the scale Ã5 ¼ ðMP m4 Þ1=5 ,
^
[email protected] Þ3
:
MP m4 (8.1) The next highest scale is Ã4 ¼ ðMP m Þ , carried by a
quartic scalar interaction, and a cubic term with a single
vector and two scalars,
3 1=4 $ ^
[email protected] Þ4
;
2
MP m6 $ ^
^
@[email protected] Þ2
:
MP m 3 Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 The next highest is a quintic scalar, and so on. The only terms
which carry a scale less than Ã3 ¼ ðMP m2 Þ1=3 are terms with
^
only scalars [email protected] Þn , and terms with one...

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