This document discusses warped dark sectors, where a dark matter particle interacts with standard model particles through a continuum of mediator states rather than a single particle. Key points: 1) A warped extra dimension with a bulk mediator field can encode a continuum/tower of Kaluza-Klein states that goes beyond the narrow width approximation at high energies. 2) The propagator for a bulk mediator field can be represented as a sum over individual KK modes or using the continuum approximation, both of which reproduce the expected fractional power potential at long distances. 3) Self-interactions in the bulk can cause high-mass KK states to have large widths and cascade decays, suppressing on-shell production but maintaining off

1. f l i p . t a n e d o @ u c r. e d u
Warped Dark Sectors
Flip Tanedo
23 March 2021
Continuum mediators
work with
Sylvain Fichet
Philippe Brax
Lexi Costantino
Kuntal Pal
Ian Chaffey
1906.02199
1910.02972
2002.12335
2102.05674
WARPED DARK SECTORS

2. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
Dark sector
2
Renormalizable
(or low-dimension)
portal interaction
one or two particles in
the narrow width limit
e.g. kinetic mixing,
operator with |H|2,
neutrino portal…
stable
Dark matter
Standard
Model
Mediator
Strength of coupling
controls abundance,
self-interactions

3. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
3
Renormalizable
(or low-dimension)
portal interaction
Continuum (or tower)
of states beyond the
narrow-width limit
e.g. kinetic mixing,
operator with |H|2,
neutrino portal…
stable
Dark matter
Standard
Model
Mediator
e.g. near-conformal sector,
“unparticles,” extra dimension…
beyond “particle” mediators

4. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
Warped Dark Sector: AdS Realization
4D dark matter, bulk mediators, new phenomena
• 5D mediator encodes Kaluza–Klein
tower (spectrum)
• Narrow width approximation breaks
down at large KK number
4
BULK
MEDIATOR
UV
BRANE
IR
BRANE
Dark matter
Standard
Model Mediator
“warping, but no attempt to address hierarchy”
5D mediator is dual to
near-conformal sector
1906.02199, 1910.02972, 2002.12335, 2102.05674
• spacelike: non-integer power potentials
• timelike: “soft bomb” cascade decays
Examples of phenomenology

5. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
Katz, Reece, Sajjad 1509.03628
“continuum mediated DM-baryon scattering”
Contino, Max, Mishra 2012.08537 “searching for elusive dark sectors”
Rizzo et al. 1801.08525, 1804.03560, 1805.08150, 2006.06858
“extra dimensions are dark”
Buyukdag, Dienes, Gherghetta, Thomas 1912.10588
“partially composite dynamical dark matter”
Bernal, Donini, Folgado, Ruis 2004.14403
“KK FIMP dark matter in warped extra dimensions”
McDonald, von Harling 1203.6646
“secluded DM coupled to a hidden CFT”
Gherghetta, von Harling 1002.2967 “a warped model of DM”
McDonald, Morrissey 1002.3361, 1010.5999
“low-energy probes of a warped extra dimension”
5
Summary of prior work
Apologies for incompleteness, please let me know of major omissions
=
1906.02199: “Warped Dark Sector”
FT, Fichet, Brax
1910.02972: “Exotic spin-dependent forces”
FT, Costantino, Fichet
2002.12335: “EFT in AdS: Continuum Regime,
Soft Bombs, …” FT, Costantino, Fichet
2102.05674: “Continuum mediated self-
interacting dark matter” FT, Chaffey, Fichet
2011.06603: “Opacity from Loops in AdS”
Costantino, Fichet
1905.05779: “Opacity and EFT” Fichet
See also: explicit RS models, explicit CFT models, unparticle mediators, hidden valley, quantum critical Higgs …

6. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
Bulk Mediator: propagator
Mixed position-momentum space
6
function for z > z0 is
(z, z0
< z) =
i⇡R
2
✓
zz0
R2
◆2
JyUV(pz0) JyIR(pz)
f
Jy(p)
(4.3)
m the method of variations, Appendix B, using the solutions to the homogeneous
u(z) = z2
J↵(pz) v(z) = z2
Y↵(pz) , (4.4)
The propagator depends on boundary functions e
J and e
Y , which are boundary
the homogeneous solutions:
e
JUV
↵ ⌘ BUV
u(z) z=R
= ( mUV + @z) u(z)|z=R (4.5)
e
JIR
↵ ⌘ w3
BIR
u(z) z=R0 = w3
(wmIR + @z) u(z) z=R0 , (4.6)
e
Jy’s are combinations of Bessel functions; poles match KK sum
Bulk self-interaction shifts
momentum into complex plane.
Higher KK modes have larger
widths (as expected) and mix
Re
Im
Fichet 1905.05779 ,Section 5: Dressed Propagator; quantitative results: Fichet & Costantino 2011.06603

7. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
Continuum-Mediated Self-Interactions
“Discretuum” Mediators from a Kaluza–Klein Tower
Closed form 5D brane-to-brane propagator matches in
fi
nite tower
of Kaluza–Klein modes: produces non-integer power-law
Cha
ff
ey, Fichet, Tanedo: 2102.05674 7
Kaluza–Klein representation. One may use the KK representation of the free
3.15) in the spectral representation of the potential (5.5); this amounts to iden
xchange of a 5D bulk scalar with the sum of t-channel diagrams with each KK m
= + + + · · ·
The spectral distribution is Disc⇢
⇥
Gp
⇢(z, z0
)
⇤
=
P
n fn(z)fn(z0
)2⇡ (p2
m2
n),
potential is an infinite sum of Yukawa potentials from each KK mode:
V (r) =
1
4⇡
2
k
X
n
fn(z0)2 e mnr
r
.
While this KK representation of V is exact, it requires knowledge of the entire s
potential is exponentially suppressed at long distances,
V (r) /
⇣m1
k
⌘1 2↵ 1
kr2
e m1r
. (5.21)
2↵, r) takes the place of the e mr
Yukawa factor that encodes the mass
mediator scenario. In turn, this mass gap is a key ingredient for cutting o↵
nge dark forces.
e to check the behavior in the gapless limit µ ! 0. The large, spacelike
ximation of the propagator (5.15) is exact in this limit and potential can
tly. We recover the gapless limit in (5.19) the gapless limit is recovered by
ving
Vgapless(r) =
2
2⇡3/2
(3/2 ↵)
(1 ↵)
1
r
✓
1
kr
◆2 2↵
, (5.22)
result from [17]. The power law behavior obtained matches the proposed
th the AdS/CFT identification = 2 ↵.
al, ↵ = 1
rameter ↵ = 1 and with generic IR brane mass parameter bIR ⇠ O(1),
ghter than the scale µ. The suppression relative to µ is the kinetic factor
2
in (5.13). This is in contrast to the ↵ < 1 case. The spectral integral
(3.15) in the spectral representation of the potential (5.5); this amounts to iden
exchange of a 5D bulk scalar with the sum of t-channel diagrams with each KK m
= + + + · · ·
The spectral distribution is Disc⇢
⇥
Gp
⇢(z, z0
)
⇤
=
P
n fn(z)fn(z0
)2⇡ (p2
m2
n),
potential is an infinite sum of Yukawa potentials from each KK mode:
V (r) =
1
4⇡
2
k
X
n
fn(z0)2 e mnr
r
.
While this KK representation of V is exact, it requires knowledge of the entire s
KK masses and wavefunctions.
Canonical representation. One may alternatively use the canonical repres
the propagator (3.13) in the spectral representation of the potential (5.5). In th
uza–Klein representation. One may use the KK representation of the free pro
5) in the spectral representation of the potential (5.5); this amounts to identif
hange of a 5D bulk scalar with the sum of t-channel diagrams with each KK mo
= + + + · · ·
spectral distribution is Disc⇢
⇥
Gp
⇢(z, z0
)
⇤
=
P
n fn(z)fn(z0
)2⇡ (p2
m2
n), so
ential is an infinite sum of Yukawa potentials from each KK mode:
V (r) =
1
4⇡
2
k
X
n
fn(z0)2 e mnr
r
.
le this KK representation of V is exact, it requires knowledge of the entire spe
In practice: mass gap from IR brane
Ian Cha
ff
ey
Our 5D EFT contains isolated degrees of freedom localized on a brane. In a realistic theory
with gravity, localized 4D fields are special modes from 5D bulk fields and are necessarily
accompanied by a spectrum of KK modes [32]. We assume an appropriate limit where the
observable e↵ects of these modes are negligible.
3.3 Model Parameters
For the purposes of studying novel, continuum-mediated dark matter self-interactions, we
restrict the parameters presented in the 5D model in Section 3.1. The AdS curvature, k,
corresponds to the cuto↵ of the theory, as described in Section 3.2. To ensure that the cuto↵
of the theory is beyond the experimental reach of the Large Hadron Collider to detect, e.g.,
Kaluza–Klein gravitons, we set k to be
k = 10 TeV . (3.9)
This sets the position of the UV brane zUV = k 1
and the upper bound on all other dimen-
sionful parameters in the theory. The AdS curvature is much smaller than the Planck scale,
in the spirit of ‘little Randall–Sundrum’ models [33].
The mediator mass, M is related to the dimension of the continuum mediator operator
and is conveniently described by the dimensionless parameter ↵,
↵2
⌘ 4 +
M2
k2
= (2 )2
. (3.10)
The range of ↵ corresponds to the branch of AdS/CFT (see details in Appendix A) and

8. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
8
Figure 2: Absolute potential |V (r)| plotted to validate the cont
Cha
ff
ey, Fichet, Tanedo: 2102.05674
5D Green’s function
Large spacelike
momentum asymptotic
Many KK modes
trust
green
line
trust
red
line
Sum of many KK modes
reproduces a fractional
power potential.
n.b. KK picture is
consistent as long as
you’re within the
effective theory (do not
probe beyond heaviest
KK mode)

9. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
9
Figure 7: Velocity dependence of the thermally averaged transfer cross section. The parameters are
chosen to be reasonably fit astronomical data. A benchmark 4D self-interacting dark matter model
Self-Interaction Phenomenology
New parameter: Bulk mass/Scaling Dimension
bulk
mass
Cha
ff
ey, Fichet, FT: 2102.05674; Visible matter long-range forces: 1910.02972 FT, Costantino, Fichet

10. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
10
AdS5 as an effective field theory
Fichet: 1905.05779; Brax, Fichet, Tanedo: 1906.02199; Costantino, Fichet, Tanedo: 2002.12335
ENERGY
SCALE
5D
fl
at (scales shorter than curvature)
5D theory breaks down
5D warped
4D EFT with contact interactions
many KK modes
few KK modes

11. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
11
Timelike momenta
High Kaluza–Klein modes have larger widths that merge into a continuum rather
than a sequence of discrete Breit–Wigner resonances.
4D Narrow width approximation breaks down (due to interactions)
See also Dynamical Dark Matter realization, Brooks, Dienes et al. 1610.04112, 1912.10588
one-“particle” states
Re
Im

12. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
12
AdS5 as an effective field theory
ENERGY
SCALE
What happens at Λμ/k ?
Fichet: 1905.05779; Brax, Fichet, Tanedo: 1906.02199; Costantino, Fichet, Tanedo: 2002.12335
5D
fl
at (scales shorter than curvature)
5D theory breaks down
5D warped
4D EFT with contact interactions
RS2-like theory (one brane)
Continuum states
RS1-like (two branes)
Discrete KK modes
IR brane “decouples”

13. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
Bulk cascades from self-interactions
Timelike mediators falling into the IR: soft bombs
But high-mass states have a large
width (continuum), so narrow width
approximation fails. Cascades are
exponentially suppressed.
See Knapen et al. 1612.00850, Cesarotti et al. 2009.08981, Hofman & Maldacena 0803.1467
13
Bulk interactions
cause decays into
lighter KK modes
Large ’t Hooft coupling
(contrast to QCD)
“soft bomb” event
Lexi Costantino

14. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
So what? Annihilation Rate
Speculations for a neat application
Bulk interaction controls the breakdown of the
narrow width approximation.
Additional control over dark matter annihilation
rate, separate from Yukawa coupling.
e.g. typical self-interacting dark matter cross section
is too large for thermal freeze out
14
cascade controlled
by bulk interaction
coupling to
dark matter
suppress on-shell (timelike)…
…keep off-shell (spacelike)

15. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
15
Lots to explore
• Cosmological Bounds
Early universe phase transition (see, e.g.
1910.10160), light particle bounds
• Stellar cooling bounds
If high-momentum mediator production
is suppressed, could this relax bounds?
Finite temperature: AdS-Schwarzschild
• Spin-1 mediator
Work with Kuntal Pal; see also recent
work by Rizzo et al. (e.g. 1801.08525)
Very much a work in progress
Brax, Fichet, Tanedo (1906.02199)
FIMP? See also
Bernal et al.
2004.14403

16. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
16
Conclusion
• 5D model for a continuum
mediator (dual to near CFT sector)
• Narrow width approximation
breaks down
• Non-integer potential
• Novel handle to play with dark
sector phenomenology
Warped dark sector
Brax, Fichet, Tanedo (1906.02199)
FIMP? See also
Bernal et al.
2004.14403
Similar to: Randall–Sundrum, CFT, unparticles,
hidden valley, quantum critical Higgs… if we
ignore the Hierarchy Problem/Higgs, focus on
phenomenology of the continuum regime.

17. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS
Looking forward to chatting in person soon
Thanks Chanda, Regina, Djuna,
Nausheen, and Tien-Tien!
Sylvain Lexi Ian
Im
age: Flip
Tanedo

18. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
18
Extra Slides in case of discussion

19. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
19
Mixed position–momentum propagator
where p2
is the norm of the Minkowski four-momentum. One may use the method of variations to
obtain the propagator with respect to the homogeneous solutions, see Appendix B:
G(z, z0
) =
i⇡
2R3
h
e
JUV
↵ z2
<Y↵(pz<) e
Y UV
↵ z2
<J↵(pz<)
i h
e
JIR
↵ z2
>Y↵(pz>) e
Y IR
↵ z2
>J↵(pz>)
i
e
JUV
↵
e
Y IR
↵
e
Y UV
↵
e
JIR
↵
, (3.9
3
Also see: https://physics.stackexchange.com/q/143601 and references therein.
4
unction for z > z0 is
(z, z0
< z) =
i⇡R
2
✓
zz0
R2
◆2
JyUV(pz0) JyIR(pz)
f
Jy(p)
(4.3)
the method of variations, Appendix B, using the solutions to the homogeneous
z> ⌘ max(z, z0
). The e
Y and e
J are shorthand for the boundary operator
solutions, for example:
e
JUV
↵ ⌘ BUV
⇥
z2
J↵(pz)
⇤
z=R
. (3.10)
e boundary conditions:
+ UV (R) = 0 BIR
(z) = ↵IR
0
(R0
) + IR (R0
) = 0 . (3.11)
of motion on the boundaries. This, in turn, depends on the Lagrangian
ecomposition
osition is
perturbing about a stable vacuum because AdS provides a positive contribu
3.1 Bulk Equation of Motion
To derive the equation of motion, we may integrate the kinetic term by part
p
g|@ |2
= @M
p
g ⇤
gMN
@N
⇤
@M
p
ggMN
@N
The bulk equation of motion is
O = @M
p
ggMN
@N
p
gM2
= 0 .
In RS the di↵erential operator maps onto:
O =
✓
R
z
◆3
"
@2
z
3
z
@z + p2
✓
R
z
◆2
M2
#
.
The integration by parts with respect to @z generates surface terms at R0
an
Z R0 Z
p
Z
p p
Solution to homogeneous equation
Boundary equation of motion
Acting on homogeneous solution
e.g. solution by method of variations

20. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
20
UV brane IR brane
elementary composite
~conformal
A dual description of a conformal sector
A slice of AdS5

21. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
21
The RS1 model
electroweak hierarchy
Randall & Sundrum hep-ph/9905221
Higgs Boson
IR brane-localized
Top quark
IR brane leaning
electron
UV brane leaning
composite
elementary
TeV-1
MPl-1
only a few
KK modes relevant
Zero mode pro
fi
les

22. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
22
Warped Dark Sector
Brax, Fichet, Tanedo: 1906.02199
UV brane
IR brane
standard
model
mediator
Goal is not the hierarchy
many KK modes!
e.g.
dark matter:
either brane

23. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
23
Cutoff is position–dependent
EFT perspective: higher-order interaction dominates for momenta of this order:
c.f. RS1 solution to the hierarchy problem
Costantino, Fichet, Tanedo: 2002.12335
Alternatively, for
fi
xed
4-momentum p, this is
a limit on the distance z
from the UV brane

24. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
24
Cascade Recursion Relation
Trick: replace KK sum with discontinuity across spectral representation
MN =
Z R0
R
du AN (u)fn(u) , (12.3)
AN (u) corresponds to the blob in the blob diagrams above. The decay rate is thus
N =
2M0
N 1
X
fin.
X
n
Z
d N (2⇡)4
Z R0
R
du AN (u)fn(u)
Z R0
R
du0
AN (u0
)⇤
fn(u0
)⇤
. (12.4)
hat we have written
N
X
fin.
=
N 1
X
fin.
X
n
, (12.5)
s ‘figurative’ notation to mean that we’ve explicitly separated the sum over the kk modes fn of
ernal state we’ve singled out. The power of this observation is that we may use the integral trick
to replace the kk sum with a spectral integral with respect to the discontinuity of the propagator
the real line:
N
X
n=n0
fn(z)fn(z0
)A(m2
n; z, z0
) =
1
2⇡
Z m2
N
m2
n0
d⇢ A(⇢; z, z0
) Disc⇢
h
p
⇢(z, z0
)
i
. (12.6)
33
1
N
X
fin.
=
N 1
X
fin.
X
n
, (12
which is ‘figurative’ notation to mean that we’ve explicitly separated the sum over the kk modes fn
the external state we’ve singled out. The power of this observation is that we may use the integral tr
(10.5) to replace the kk sum with a spectral integral with respect to the discontinuity of the propaga
across the real line:
N
X
n=n0
fn(z)fn(z0
)A(m2
n; z, z0
) =
1
2⇡
Z m2
N
m2
n0
d⇢ A(⇢; z, z0
) Disc⇢
h
p
⇢(z, z0
)
i
. (12
33
X(z, z0
) =
N
X
n=n0
fn(z)fn(z0
)A(m2
n; z, z0
) .
We can convert the sum over kk wavefunctions into a contour integral over the magnitude of
momentum:
X(z, z0
) =
1
2⇡
I
C
d⇢ Gp
⇢(z, z0
)A(⇢; z, z0
) .
The contour C is a series of counter-clockwise loops around the n0 through Nth poles. This is man
true with the kk decomposition of G(z, z0), but the real power is that one may replace it with t
position-space propagator. Furthermore, one may deform the series of loops into a single counter-cloc
contour that encloses the desired poles:
=)
29
Costantino, Fichet, Tanedo: 2002.12335

25. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
25
Cascade Recursion Relation
See also, e.g. treatment of unstable particles in (review): Phys.Part.Nucl. 45 (2014), Kuksa “Unstable States in QFT”
cascade decay composed of 1 ! 2 branchings coming from a
an initial state is
⇧N |MN |2
⌘
1
2M0
Z
(2⇡)4
d N |MN |2
, (12.1)
factor and M0 is the [would-be] mass of the initial excitation.
spect to d N , the phase space element defined in the pdg that
e these cascades, my collaborators Sylvain and Lexi developed
that relates an N-final state cascade to an (N + 1)-final state
ombs
n 5 of [7]. Assume a cascade decay composed of 1 ! 2 branchings coming from a
. The decay rate for an initial state is
N =
1
2M0
Z
d⇧N |MN |2
⌘
1
2M0
Z
(2⇡)4
d N |MN |2
, (12.1)
propriate phase space factor and M0 is the [would-be] mass of the initial excitation.
write this out with respect to d N , the phase space element defined in the pdg that
In order to investigate these cascades, my collaborators Sylvain and Lexi developed
te recursion relation that relates an N-final state cascade to an (N + 1)-final state
cascade:
The decay rate for an N-final state cascade is
N =
1
2M0
N
X
fin.
Z
d N (2⇡)4
|MN |2
,
where d N is the N-body phase space factor that contains the overall momentum-conservin
The sum is over the distinguishable configurations of the N final states. A convenient sh
write an N particle amplitude for a particular final state with respect to its ‘last’ vertex at
ay rate for an N-final state cascade is
N =
1
2M0
N
X
fin.
Z
d N (2⇡)4
|MN |2
, (12.2)
N is the N-body phase space factor that contains the overall momentum-conserving -function.
m is over the distinguishable configurations of the N final states. A convenient shorthand is to
N particle amplitude for a particular final state with respect to its ‘last’ vertex at z = u,:
MN =
Z R0
R
du AN (u)fn(u) , (12.3)
N (u) corresponds to the blob in the blob diagrams above. The decay rate is thus
N =
2M0
N 1
X
fin.
X
n
Z
d N (2⇡)4
Z R0
R
du AN (u)fn(u)
Z R0
R
du0
AN (u0
)⇤
fn(u0
)⇤
. (12.4)
at we have written
N
X
fin.
=
N 1
X
fin.
X
n
, (12.5)
‘figurative’ notation to mean that we’ve explicitly separated the sum over the kk modes fn of
he decay rate for an N-final state cascade is
N =
1
2M0
N
X
fin.
Z
d N (2⇡)4
|MN |2
,
here d N is the N-body phase space factor that contains the overall momentum-conserving
he sum is over the distinguishable configurations of the N final states. A convenient shor
ite an N particle amplitude for a particular final state with respect to its ‘last’ vertex at z
MN =
Z R0
R
du AN (u)fn(u) ,
here AN (u) corresponds to the blob in the blob diagrams above. The decay rate is thus
Z Z Z
Distinguishable
Final states
1

26. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
26
Cascade Recursion Relation
rate for an N-final state cascade is
N =
1
2M0
N
X
fin.
Z
d N (2⇡)4
|MN |2
, (12.
is the N-body phase space factor that contains the overall momentum-conserving -functio
s over the distinguishable configurations of the N final states. A convenient shorthand is
at in the propagators we have written p = p
p2 .15 For a scalar decay there is no angular
nce in the two-body rate. We thus have
R
d⌦mn = 4⇡ so that
N+1 =
2
2M0
N 1
X
fin.
Z
d N
Z
dp2
m dp2
n
Z
d2
u d2
v
Z
dq2
1/2(q2, p2
1, p2
2)
64⇡4q2
A(u)A(u0
)⇤
⇥
✓
R2
vv0
◆5
q(u, v) q(u0
, v0
)⇤
Discp2
m
⇥
pm (v, v0
)
⇤
Discp2
n
⇥
pn (v, v0
)
⇤
. (12.16)
AdS/CFT
that the continua tend to decay near kinematic threshold. The cascades gives rise to soft spherical
final states, in accordance with former results from both gravity and CFT sides.
Integrating over p2
1, p2
2, v, and v0
, we have
PM+1 =C↵
X
FS(M 1)
(2⇡)4
Z
d M
Z
dq2
k
⇣q
k
⌘2↵
Z
du
Z
du0
IM (u) I⇤
M (u0
)(ku)2+↵
(ku0
)2+↵
, (5.11)
where the constant prefactor is
C↵ =
84(1 ↵) 2
↵4⇡4k
✓
(1 ↵) sin(⇡↵)
(1 + ↵)
◆2 |(2 + 3↵)4↵
(↵ + 2) (1 ↵)
(1+↵)
ei↵⇡
|2
(2 + 3↵)2(2 + ↵)2(1 + ↵)2
. (5.12)
One may replace the dq2
in favor of a sum over the continuum of KK final states by applying (5.5).
This yields a recursion relation
PM+1 = r
Z X
FS(M)
Z
du IM (u)fn(u)
2
(2⇡)4
d n = r PM . (5.13)
The fact that one obtains a simple relation is a consequence of the integrand having a specific
momentum dependence and is nontrivial. This relation is clearly useful since it can be used to
give an estimate of a total rate with arbitrary number of legs.
The recursion coefficient r is given by
r ⌘
2
k
1
10241+↵
1
2⇡3↵3
|(2 + 3↵)4↵
(2 + ↵) (1 ↵)
(1+↵)
ei↵⇡
|2
(2 + 3↵)2(2 + ↵)2(1 + ↵)2
!
(1 ↵) sin(⇡↵)
(1 + ↵)
. (5.14)
Even for the strongly coupled case, 2
⇠ `5k, this coefficient is much smaller than one.
small
Källén triangle function

27. f l i p . t a n e d o @ u c r. e d u WARPED DARK SECTORS 16
27
Cascade exponential suppression
Opacity at work
ENERGY
SCALE
5D warped continuum
RS2-like theory (one brane)
Continuum states
5D warped KK regime
RS1-like (two branes)
Discrete KK modes
27
Costantino, Fichet, Tanedo: 2002.12335