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**Unformatted text preview: **vector and the rest
^
^
scalars @Að@2 Þn .
The scale Ã3 is carried only by the following terms:
$ ^
^
hð@2 Þn
;
nÀ1 2nÀ2
MP m $ ^
^
ð@AÞ2 ð@2 Þn
:
n 2n
MP m (8.3) All other terms carry scales higher than Ã3 .
It turns out that we can arrange to cancel all of the scalar
self-couplings by appropriately choosing the coefﬁcients of
the higher order terms. We work with the form of the potential
in Eq. (5.9) where indices are raised with the full metric, and
¨
the Stuckelberg formalism of Sec. VI.B. We do so because we
eventually want to keep track of powers of h, so the form of
¨
the Stuckelberg replacement in Sec. VI.B is simpler. We are
interested only in scalar self-interactions, so we may make the
replacement (6.31) with the vector ﬁeld set to zero,
H ! 2@ @ À @ @ @ @ : (8.4) The interaction terms are a function of the matrix of second
derivatives Å @ @ . As reviewed in the Appendix,
there is at each order in a single polynomial in Å which
is a total derivative. By choosing the coefﬁcients (5.9) correctly, we can arrange for the terms to appear in these total
derivative combinations. The total derivative combinations
have at each order in as many terms as there are terms in
the potential of Eq. (5.9), so all the coefﬁcients must be ﬁxed,
except for one at each order which becomes the overall
coefﬁcient of the total derivative combination.
The choice of coefﬁcients in the potential (5.9) which
removes the scalar self-interactions is, to ﬁfth order (de
Rham and Gabadadze, 2010a),
c1 ¼ 2 c 3 þ 1 ;
2 c2 ¼ À3c3 À 1;
2 (8.5) 1
d1 ¼ À6d5 þ 16ð24c3 þ 5Þ; d2 ¼ 8 d 5 À 1 ð 6 c 3 þ 1 Þ ;
4
d3 ¼
d4 ¼ 1
3d5 À 16ð12c3
À6d5 þ 3c3 ;
4 (8.6) þ 1Þ; 7
f1 ¼ 32 þ 9c3 À 6d5 þ 24f7 ;
8
5
f2 ¼ À32 À 15c3 þ 6d5 À 30f7 ;
16 A. Tuning interactions to raise the cutoff $ 697 (8.2) f3 ¼ 3c3 À 3d5 þ 20f7 ;
8
1
f4 ¼ À16 À 3c3 þ 5d5 À 20f7 ;
4 (8.7) 3
f5 ¼ 16c3 À 3d5 þ 15f7 ; f6 ¼ d5 À 10f7 :
At each order, there is a one-parameter family of choices that
works to create a total derivative. Here c3 , d5 , and f7 are
chosen to carry that parameter at order 3, 4, and 5, respectively. Note, however, that at order 5 and above (or D þ 1 and
above if we were doing this in D dimensions), there is one
linear combination of all the terms, the characteristic polynomial of h mentioned below Eq. (5.8) that vanishes identically. This means that one of the coefﬁcients is redundant, Kurt Hinterbichler: Theoretical aspects of massive gravity 698 and we can, in fact, set d5 and its higher counterparts to any
value we like without changing the theory. Thus there is only
a two parameter family (D À 2 parameter in dimension D) of
theories with no scalar self-interactions. This can be carried
through at all orders, and at the end there will be no terms
$ð@2 Þn .
The only terms with interaction scales lower than Ã3 were
^
the scalar self-interactions ð@2 Þn , and terms with one vector...

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