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**Unformatted text preview: **coefﬁcients
C1 , C2 , etc.
¨
VII. STUCKELBERG ANALYSIS OF INTERACTING
MASSIVE GRAVITY ¨
In this section, we set D ¼ 4 and apply the Stuckelberg
analysis to the massive GR action (5.1) in the case of a ﬂat
absolute metric. The mass term reads
Smass M2 m2 Z 4 ¼À P
d x ðh h À h h Þ:
24
(7.1) ¨
The Stuckelberg analysis instructs us to make the replacement (6.18),
h ! H
¼ h þ @ A þ @ A þ @ A @ A þ 2@ @
þ @ @ @ @ Á Á Á : (7.2) The extra terms with h in the ellipsis will not be important for
this theory, as we will see.
At the linear level, this replacement is exactly the linear
¨
Stuckelberg expansion of Sec. IV. We have to canonically
normalize the ﬁelds here to match the ﬁelds of the linear
analysis. Using a hat to signify the canonically normalized
ﬁelds with the same coefﬁcients as used in Sec. IV (although
there we omitted the hats), we have
^
h ¼ 1MP h;
2 ^
A ¼ 1mMP A;
2 ^
¼ 1m2 MP :
2
(7.3) $ ^
[email protected] Þ3
;
Ã5
5 693 Ã5 ¼ ðMP m4 Þ1=5 : (7.6) In terms of the canonically normalized ﬁelds (7.3), the
gauge symmetries (6.19) read
2
^
^
^
h ¼ @ þ @ þ
L ^h ;
MP
2 ^ ^
^
^
^
A ¼ @ Ã À m þ
@ A
MP
2 ^ ^ ^
A A @ @ À Á Á Á ;
À
2
mMP
^
¼ ÀmÃ; (7.7) ^
^
where we rescaled Ã ¼ ðmMP =2ÞÃ and ¼ ðMP =2Þ .
Finally, note that since the scalar ﬁeld always appears
¨
with at least two derivatives in the Stuckelberg replacement
(7.2), the resulting action is automatically invariant under the
global Galilean symmetry
ðxÞ ! c þ b x ; (7.8) where c and b are constants. In addition, the action is
automatically invariant under global shifts in A !
A þ c for constant c . It will persist even in limits where
the gauge symmetries on A and no longer act.
A. Decoupling limit and breakdown of linearity As seen in Sec. IV.B, the propagators have all been made to
go as $1=p2 , so normal power counting applies, and the
lowest scale Ã5 is the cutoff of the effective ﬁeld theory. To
focus in on the cutoff scale, we take the decoupling limit
m ! 0; MP ! 1 ; T ! 1; Ã5 ; T
fixed:
MP
(7.9) (7.5) All interaction terms go to zero, except for the scalar cubic
term (7.6) responsible for the strong coupling, which
we calculate using the replacement H ¼ 2@ @ þ
@ @ @ @ since we do not need the vector and tensor
terms. As discussed in Sec. IV.B, we must also do the
conformal transformation h ¼ h0 þ m2 . This will
diagonalize all the kinetic terms (except for various cross
terms proportional to m which are eliminated with appropriate gauge ﬁxing terms, as discussed in Sec. IV.B, and which
go to zero anyway in the decoupling limit).
After all this, the Lagrangian for the scalar reads, up to a
total derivative,
Z
2
^
^
^
^
S ¼ d4 x À 3ð@Þ2 þ 5 ½ðhÞ3 À ðhÞð@ @ Þ2
Ã5
1^
þ
T:
(7.10)
MP The larger , the smaller the scale, since m < MP . We have
n þ nA þ nh ! 3, since we are only considering int...

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