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**Unformatted text preview: **Nicolis and Rattazzi, 2004;
Gabadadze and Iglesias, 2006) and are suppressed by the
scale Ã3 ¼ ðM4 m2 Þ1=3 (in fact, this was where the Galileons
were ﬁrst uncovered). In this sense, DGP is analogous to the
nicer Ã3 theories of Sec. VIII. The theory is free of ghosts and
instabilities around solutions connected to ﬂat space (Nicolis
and Rattazzi, 2004), but changing the asymptotics to the selfaccelerating de Sitter brane solutions ﬂips the sign of the
kinetic term of the longitudinal mode, so there is a massless
ghost around the self-accelerating branch (Koyama, 2007).
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 C. Resonance gravitons Ài pﬃﬃﬃﬃﬃﬃ :
p þ m p2 (9.51) 2 Setting z ¼ Àp2 , the propagator has a branch cut in the z
plane from ð0; 1Þ, with discontinuity
2m
À pﬃﬃﬃ
:
zð z þ m 2 Þ (9.52) A branch cut can be thought of as a string of simple poles,
in the limit where the spacing between the poles and
their
R1 residues both go to zero. The function fðzÞ ¼
À1 dðÞ=ðz À Þ has a cut along the real axis everywhere
that is nonzero, with discontinuity À2iðzÞ. We can see
this by noting
Z1
1
1
À
disc fðzÞ ¼
dðÞ
z À þ i z À À i
À1
Z1
¼
dðÞ½À2iðz À Þ:
À1 Using all this, and the fact that analytic functions are
determined by their poles and cuts, we can write the propagator in the spectral form
Z1
Ài
Ài
ds 2
ðsÞ;
pﬃﬃﬃﬃﬃﬃ ¼
2 þ m p2
p þs
0
p
m
ðsÞ ¼ pﬃﬃﬃ
> 0:
(9.53)
sð s þ m 2 Þ
The spectral function is greater than zero, so this theory
contains a continuum of ordinary (nonghost, nontachyon)
gravitons, with masses ranging from 0 to 1. This is what
would be expected from dimensionally reducing a noncompact ﬁfth dimension. The Kaluza-Klein tower has collapsed
down into a Kaluza-Klein continuum.
In the limit m ! 0, where the action becomes purely four
dimensional, the spectral function reduces to a delta function,
ðsÞ ! 2ðsÞ; (9.54)
Ài=p2 representing a single
and the propagator reduces to
massless graviton, vector, and scalar, as can be seen from Kurt Hinterbichler: Theoretical aspects of massive gravity 706 Eq. (4.37) (the extra factor of 2 is taken care of by noting that
the integral is from 0 to 1, so only half of the delta function
actually gets counted). This theory therefore contains a vDVZ
discontinuity.
The potential of a point source of mass M sourced by
this resonance graviton displays an interesting crossover
behavior. Looking back at pﬃﬃﬃﬃﬃﬃ (3.11) with the momentum
Eq.
space replacement m2 ! m p2 , and using the relation ¼
Àh00 =M4 for the Newtonian potential, we have
À 2 M Z d 3 p i p Áx
À1
e
2
3
2 þ mjpj
3M 4
ð 2 Þ
p
2
M
r
r
ci
¼
sin
2
3 M 4 2 2 r
r0
r0
1
r
r
þ cos
À 2si
;
2
r0
r0 ðrÞ ¼ (9.55)
X. CONCLUSIONS AND FUTURE DIRECTIONS where
siðxÞ Z x dt
0 t sint; ciðxÞ þ lnx þ Z x dt
0 t (9.56)
ðcost À 1Þ; % 0:577 . . . is the Euler-Mascer...

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