JHEP09(2018)095
Using now the first expression in eq. (2.11) we get
φ
0
0
(r) =
1
2
W
0
0
(φ
0
), φ
0
1
(r) =
1
2
φ
1
W
00
0
(φ
0
) + W
0
1
(φ
0
)
, (3.15)
φ
1
(r) ≡ φ
1
[φ
0
(r)] = W
0
0
(φ
0
)
Z
φ
0
C
1
W
0
1
(
¯
φ)
[W
0
0
(
¯
φ)]
2
d
¯
φ , (3.16)
where eq. (3.16) defines the field φ
1
(r), while the first relation in eq. (3.15) is the usual
equation for φ
0
(r) [cf. eq. (2.11)]. The integration constants have been chosen to fulfill
the BCs
φ(0) = v
0
, φ(r
1
) = v
1
, (3.17)
corresponding to the values of φ(r) in the UV and IR branes, respectively. In particular
one can fix C
1
= v
0
such that φ
0
(0) = v
0
and φ
1
(0) = 0. Then the condition φ(r
1
) = v
1
leads to fixing the integration constant s as
5
s(r
1
) =
v
1
− φ
0
(r
1
)
φ
1
[φ
0
(r
1
)]
. (3.18)
Therefore the superpotential in eq. (3.14) gets an explicit dependence on the brane distance,
W (r
1
), which in turn creates a non-trivial dependence on r
1
of the effective potential of
eq. (3.6). As the latter only gets contributions from the branes, one can then expand the
superpotential on the branes as
W (v
α
) = W
0
(v
α
) + s(r
1
)W
1
(v
α
) (3.19)
so that the effective potential can be expanded to first order in s(r
1
):
V
eff
(r
1
) = Λ
0
− W
0
(v
0
) (3.20)
+ e
−4A
0
(r
1
)
n
[Λ
1
+ W
0
(v
1
)] [1 − 4A
1
s(r
1
)] + s(r
1
)
h
W
1
(v
1
) − e
4A
0
(r
1
)
W
1
(v
0
)
io
.
Eq. (3.20) involves several key parameters that play a relevant role in our analysis.
The second line, and in particular the function s(r
1
), provides a non-trivial dependence on
the brane distance r
1
. We anticipate that r
1
can be interpreted as the constant background
value of the (canonically unnormalized) radion/dilaton field. Consequently, the cosmolog-
ical constant at the minimum of the radion potential can be set to zero by an accurate
choice of the terms in the first line, which are independent of r
1
. We fine-tune Λ
0
for such
a purpose.
6
Similarly, from eq. (3.14) and the second expression in eq. (2.11) one finds
A
0
0
(r) =
κ
2
6
W
0
(φ
0
) ,
A
0
1
(r) =
κ
2
6
φ
1
W
0
0
(φ
0
) + W
1
(φ
0
)
. (3.21)
5
For the case of finite γ
α
, eq. (3.18) has O(1/γ
α
) corrections.
6
This one is the cosmological constant fine-tuning of the theory.
– 8 –
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