JHEP04(2020)036
The action in 5D is:
S = S
gravity
+ S
IR
+ S
UV
(2.6)
where the gravitational part is, in the Jordan frame:
S
gravity
=
M
3
5
2
Z
d
4
x
Z
π
0
r
c
dy
p
G
(5)
e
S
h
R
(5)
+ G
MN
(5)
∂
M
S∂
N
S + 4k
2
i
, (2.7)
with G
(5)
MN
and R
(5)
the 5-dimensional metric and Ricci scalar, respectively, and S the
(dimensionless) dilaton field, S = 2kr
c
|y|. We consider for the two brane actions the
following expressions:
S
IR
=
Z
d
4
x
q
−g
(4)
IR
e
S
−f
4
IR
+ L
SM
+ L
DM
(2.8)
and
S
UV
=
Z
d
4
x
q
−g
(4)
UV
e
S
−f
4
UV
+ . . .
, (2.9)
where f
IR
, f
UV
are the brane tensions for the two branes and g
(4)
IR,UV
= −G
(5)
/G
(5)
55
is the
determinant of the induced metric on the IR- and UV-brane, respectively. Throughout
the paper, we consider all the SM and DM fields localized on the IR-brane, whereas on
the UV-brane we could have any other physics that is Planck-suppressed. We assume that
DM particles only interact with the SM particles gravitationally by considering only DM
singlets under the SM gauge group. More complicated DM spectra with several particles
will also not be studied here.
Notice that the gravitational action is not in its canonical form. Going to the Einstein
frame changing G
(5)
MN
→ exp(−2/3S)G
(5)
MN
, we get:
S
gravity
=
Z
d
4
x
Z
π
0
r
c
dy
p
−G
(5)
M
3
5
2
R
(5)
−
1
3
G
MN
(5)
∂
M
S∂
N
S + 4e
−
2
3
S
k
2
(2.10)
+
Z
d
4
x
Z
π
0
r
c
dy
q
−g
(4)
e
−
S
3
δ(y − y
0
)
−f
4
IR
+ L
SM
+ L
DM
− δ(y − π)f
4
UV
,
where now the gravitational action is the Einstein action and from the kinetic term of the
dilaton field we can read out that the physical field must be rescaled as
M
3/2
5
/
√
3
S.
Eventually, it is important to stress that, in the Einstein frame, the brane action terms
still have an exponential dependence e
−S/3
from the dilaton field. This action has a shift
symmetry S → S + const in the limit k → 0, that makes a small value of k with respect
to M
5
“technically natural” in the ’t Hooft sense. Using the action above in the Einstein
frame, it can be shown that the metric in eq. (2.3) can be recovered as a classical background
if the brane tensions are chosen as:
f
4
IR
= −f
4
UV
= −4k M
3
5
. (2.11)
Notice that, in a pure 4-dimensional scenario, the gravitational interactions would be
enormously suppressed by powers of the Planck mass, while in an extra-dimensional one
– 7 –
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