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**Unformatted text preview: **khalava (2011).
E. Quantum corrections in the Ã3 theory As in Sec. VII.D, we expect quantum mechanically the
presence of all operators with at least two derivatives per ,
now suppressed by the cutoff Ã3 (we ignore for simplicity the
scalar-tensor interactions),
$ ^
@q ð@2 Þp
Ã3pþqÀ4
3 : (8.32) These are in addition to the classical Galileon terms in
Eq. (8.18), which have fewer derivatives per and are of
the form
$ ^
^
ð@Þ2 ð@2 Þp
Ã3p
3 : (8.33) Kurt Hinterbichler: Theoretical aspects of massive gravity An analysis similar to that of Sec. VII.D shows that the terms
(8.32) become important relative to the kinetic term at the
radius r $ ðM=MPl Þ1=3 ð1=Ã3 Þ. This is the same radius at
which classical nonlinear effects due to Eq. (8.33) become
important and alter the solution from its Coulomb form. Thus
we must instead compare the terms (8.32) to the classical
nonlinear Galileon terms (8.33). We see that the terms (8.32)
are all suppressed relative to the Galileon terms (8.33) by
powers of @=Ã3 , which is $1=Ã3 r regardless of the nonlinear
solution. Thus, quantum effects do not become important
until the radius
1
;
(8.34)
rQ $
Ã3
which is parametrically smaller than the Vainshtein radius
(8.22).
This behavior is much improved from that of the Ã5 theory,
in which the Vainshtein region was swamped by quantum
correction. Here there is a parametrically large intermediate
classical region in which nonlinearities are important but
quantum effects are not, and in which the Vainshtein mechanism should screen the extra scalar. In this region, GR should
be a good approximation; see Fig. 2.
As in the Ã5 theory, quantum corrections are generically
expected to ruin the various classical tunings for the coefﬁcients, but the tunings are still technically natural because the
corrections are parametrically small. For example, cutting off
^
loops by Ã3 , we generate the operator $ð1=Ã2 ÞðhÞ2 , which
3
^
corrects the mass term. The canonically normalized is
related to the original dimensionless metric by h $
^
ð1=Ã3 Þ@@, so the generated term corresponds in unitary
3
2
gauge to Ã4 h2 ¼ Mp m2 ðÃ3 =Mp Þh2 , representing a mass
3
2 $ m2 ðÃ =M Þ. This mass correction is paracorrection m
3
p
metrically smaller than the mass itself and so the hierarchy
m ( Ã3 is technically natural. This correction also ruins the
Fierz-Pauli tuning, but the pathology associated with the
detuning of Fierz-Pauli, the ghost mass, is m2 $
g
m2 =ðm2 =m2 Þ $ Ã2 , safely at the cutoff.
3
We mention another potential issue with the Ã3 theory. It
was found by Nicolis, Rattazzi, and Trincherini (2008) that
Lagrangians of the Galileon type inevitably have superluminal propagation around spherical background solutions. No
matter what the choice of parameters in the Lagrangian, if the
solution is stable, then superluminality is always present at
distances far enough from the source [see also Osipov and 701 Rubakov (2008)]. It was argued that such s...

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