**Unformatted text preview: **tives per ﬁeld, the so-called Galileon terms (Nicolis,
Rattazzi, and Trincherini, 2008) which change by a total derivative
under the Galileon symmetry (Nicolis, Rattazzi, and Trincherini,
2008). However, there is a nonrenormalization theorem that says
these are not generated at any loop by quantum corrections
(Hinterbichler, Trodden, and Wesley, 2010), so we need not include
them. We encounter them later when we raise the cutoff to Ã3 . Kurt Hinterbichler: Theoretical aspects of massive gravity 696 To compare with Eq. (7.19), we go back to the original
^
normalization for the ﬁelds by replacing $ m2 MP and
recall that @ @ comes from an h to ﬁnd that in unitary
gauge the coefﬁcients cp;q are similar to
À
p
2
cp;q $ Ã5 3pÀqþ4 MP m2p ¼ ðm16À4qÀ2p MPpÀqþ4 Þ1=5 : (7.21)
This comparison is possible because the operations of taking
the decoupling limit and computing quantum corrections
should commute.
Notice that the term with p ¼ 2, q ¼ 0 is a mass term,
2
$ðMP m4 =Ã2 Þh2 , corresponding to a mass correction m2 ¼
5
2
m ðm2 =Ã2 Þ. This is down by a factor of m2 =Ã2 from the tree
5
5
level mass term. Thus a small mass graviton m ( Ã5 is
technically natural, and there is no quantum hierarchy problem associated with a small mass. This is in line with the
general rule of thumb that a small term is technically natural
if a symmetry emerges as the term is dialed to zero. In this
case, it is the diffeomorphism symmetry of GR which is
restored as the mass term goes to zero. The quantum mass
correction will also generically ruin the Fierz-Pauli tuning,
but its coefﬁcient is small enough that ghosts and tachyons
associated with the tuning violation are postponed to the
cutoff; indeed the resulting ghost mass, using the relations
in Sec. II, is $Ã5 .
It is important that there are no nonparametric modiﬁcations to the kinetic structure of the Einstein-Hilbert term, even
though the lack of gauge symmetry suggests that we are free
to make such modiﬁcations. Suppose we try to calculate the
mass correction directly in unitary gauge. The graviton mass
term contributes no vertices but alters the propagator so that
its high energy behavior is $k2 =m4 (the next leading terms
are similar to 1=m2 and then 1=k2 ). At one loop, there are
two one-particle irreducible diagrams correcting the mass:
^
one containing two cubic vertices ð1=MP [email protected] h3 from the
Einstein-Hilbert action and two propagators, and another
2
^
containing a single quartic vertex ð1=MP [email protected] h4 from the
Einstein-Hilbert action and a single propagator. Cutting off
the loop at the momenta kmax $ Ã5 , the ﬁrst diagram gives
2
the largest naive correction m2 $ ð1=MP m8 ÞÃ12 $ Ã2 . (The
5
5
second diagram gives a smaller correction.) This is at the
cutoff, dangerously higher than the small correction m2 $
¨
m2 ðm2 =Ã2 Þ we found in the Stuckelberg formalism.
5
This means that there must be a nontrivial cancellation of
this leading divergence in unitary g...

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