Warped Dark Sector

f l i p . t a n e d o @ u c r. e d u KIAS HEP-PH SEMINAR
Warped Dark Sector
Flip Tanedo
Nov 3, 2020
Near-conformal mediators
1906.02199, 1910.02972, 2002.12335 …
work with
Sylvain Fichet
Philippe Brax
Lexi Costantino
Kuntal Pal
Ian Chaffey

f l i p . t a n e d o @ u c r. e d u KIAS HEP-PH SEMINAR 49
Warped dark sector
New phenomena with bulk mediators
1906.02199, 1910.02972, 2002.12335 …
2
• Breakdown of narrow width
approximation at high KK number

• Application to dark matter:  
bulk mediator with low KK scale
• e.g. Soft bomb suppression

• e.g. fractional power potential

• Possibilities for rich
phenomenology?

Brief summary of prior work by others
Apologies for incompleteness, please let me know of major omissions
• 0801.4015 Hebecker & von Harling “Sequestered Dark Matter”
• 1002.2967 Gherghetta & von Harling “A warped model of dark matter”
• 1203.6646 von Harling & McDonald “Secluded DM, Hidden CFT”
• 1912.10588 Buyukdag “Partially Composite Dynamical DM”
• 2004.14403 Bernal et al. “KK FIMP DM in Warped XD”
• 2006.01840 Betzios et al. “Global Sym., hidden sectors & emergent interactions”
+ unparticles (e.g. 0902.3676), Hidden Valley manifestations
3
1905.05779 Sylvain Fichet “Opacity and Eﬀective Field Theory in AdS”
theoretical framework

Outline
4
Review: dark sectors
Review: 5D
opacity vs.
narrow widths
cascade decays
self-interactions
Chris Burden, Urban Light, 2008
LedCrafter: etsy.com/listing/686581274

Outline
Lamp posts
5

Chris Burden, Urban Light, 2008, Los Angeles County Museum of Art; photo courtesy of @neohumanity via Instagram
Lamp posts for dark matter models (many of them)

Axion
All sorts of dark matter candidates
… we are ignoring several classes,
including some very well motivated &
exciting scenarios

WIMP
Weakly-interacting massive particle
Direct interaction with ordinary matter;
typically byproduct of theories of
electroweak naturalness
Template example: neutralino
Status: endangered species

Dark
Sectors
Dark matter + mediator
Dark matter is totally neutral.
Interacts through a light mediator particle
Assumption: single (few) mediator model
captures phenomenology of many explicit
theories.
Standard Model
Mediator
Dark Matter

See, e.g. Dark Sectors 2016 1608.08632
10
e
e
e
e
e
e
e
e
e e
capture
a
n
n
i
h
i
l
a
t
i
o
n
A0
A0
INDIRECT DIRECT PRODUCTION
SELF
Standard Model
Mediator
Dark Matter
SM
SM
SM SM

Dark
Sector
Dark matter + mediator
Interacts through a light mediator particle
Assumption: single (few) mediator model
captures phenomenology of many explicit
theories.
Standard Model
Mediator
Dark Matter

5D Dark
Sector
Dark matter + 5D mediator
Interacts through a bulk mediator particle
Assumption: behavior of many KK modes
is qualitatively different from that of a few
mediators.
… not obvious from our experiences with
standard 5D BSM model building!
Standard Model
5D Mediator
Dark Matter
LedCrafter:
etsy.com/listing/686581274

Outline
Lessons from the fifth dimension
13

14
UV brane IR brane
elementary composite
~conformal
A dual description of a conformal sector
A slice of AdS5

The RS1 model
electroweak hierarchy
Randall & Sundrum hep-ph/9905221
15
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 profiles

Warped Dark Sector
Brax, Fichet, Tanedo: 1906.02199
16
UV brane
IR brane
standard
model
mediator
?
Goal is not the hierarchy
many KK modes!
e.g.
dark matter:
either brane

Outline
Opacity in the fifth dimension
17

AdS5 as an effective field theory
Fichet: 1905.05779; Brax, Fichet, Tanedo: 1906.02199; Costantino, Fichet, Tanedo: 2002.12335
18
ENERGY
SCA
E
5D flat (scales shorter than curvature)
5D theory breaks down
5D warped
4D EFT with contact interactions
many KK modes
few KK modes
is also warped down
See work by Sylvain Fichet
I will focus on 5D picture

Opacity: cutoff is position–dependent
c.f. RS1 solution to the hierarchy problem
EFT perspective: higher-order interaction dominates for momenta of this order:
Costantino, Fichet, Tanedo: 2002.12335
19
Alternatively, for fixed
4-momentum p, this is
a limit on the distance z
from the UV brane

Obvious for space-like momentum
Space-like momentum exchange related to range of potential
20
e 1: t-channel diagrams generating long-range forces in the
sent strong dynamics.
Max 4-momentum for
UV brane correlation function
to reach IR brane z~1/μ

ENERGY
SCA
E
What happens at Λμ/k ?
21
5D warped
RS2-like theory (one brane)
Continuum states
RS1-like (two branes)
Discrete KK modes
IR brane “decouples”

Not obvious: true for time-like momentum?
What happens to high-momentum, on-shell probes?
Brax, Fichet, Tanedo: 1906.02199
22
?
Puzzle: high-p on-shell particle
produced in the UV.
If it reaches the IR, then it
exceeds the theory’s cutoff.
What happens to it?

Possible loophole: maybe it decays?
23
proposal: maybe you end up
with a cascade of lower-mass
particles.
By splitting energy between
many final states, you stay
below the z-dependent cutoff.

Mechanism: bulk field self-interactions
Assume ɸ3 bulk interaction (model for any non-self-interaction)
causes ɸ to decay into lower
Kaluza–Klein modes
Fichet: 1905.05779, Costantino, Fichet, Tanedo: 2002.12335
24
Consistent with mɸ ≪ Λ
5D loop factor
imaginary contribution to 1PI self
energy at loop-level

Outline
Cascade decays: opacity for timelike four-momenta
25

“Soft Bomb”
Model: bulk scalar mediator
26
BULK
MEDI
U
B
E
I
B
E
erms. Here M2
is a bulk mass parameter that will control the
articularly important parameter that shows up in the following
↵2
⌘ M2
R2
+ 4 . (3.2)
[ ] = 3/2. The BF bound states that ↵2
0. See (4.7) of
of this bound is that one may have negative bulk mass2
while
ecause AdS provides a positive contribution to the energy.
otion
e may integrate the kinetic term by parts.
p ⇤ MN ⇤ p MN
Conformal scaling dimension: ɑ=Δ-2
Bulk mass parameter
RmUV + (2 ↵) ) mUV = k [(2 ↵) bUV] . (3.30)
ventions. Sylvain writes the UV brane mass with the opposite sign so that
nd bSyl.
UV = bUV. As a result, the boundary mass terms take the form:
yl. = k
h⇣
2 ↵ + bSyl.
UV
⌘
(z R) + (↵ 2 + bIR) (z R0
)
i
| |2
. (3.31)
my boundary mass terms are
L =
1
R
[(2 ↵ bUV) (z R) + (↵ 2 + bIR) (z R0
)] | |2
. (3.32)
6
bIR ⌘ RmIR + (2 ↵) ) mIR = k [ (2 ↵) + bIR] .
This is a standard definition, but is not often explicitly derived. Analogously for the u
0 = c1 [ mUV + @z] z2
J↵(mnz) + c2(· · · )
= c1 [( RmUV + (2 ↵)) RJ↵(mnz) + mnRJ↵ 1(mnz)] + c2(· · · )
= c1 [bUVRJ↵(mnz) + mnRJ↵ 1(mnz)] + c2(· · · ),
where we have defined the uv brane dimensionless bulk mass parameter
bUV = RmUV + (2 ↵) ) mUV = k [(2 ↵) bUV] .
A note on conventions. Sylvain writes the UV brane mass with the opposite s
MSyl.
UV = mUV and bSyl.
UV = bUV. As a result, the boundary mass terms take the form
LSyl. = k
h⇣
2 ↵ + bSyl.
UV
⌘
(z R) + (↵ 2 + bIR) (z R0
)
i
| |2
.
By comparison, my boundary mass terms are
1 0 2
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
u(z) = z2
J↵(pz) v(z) = z2
Y↵(pz) , (4.4)
he propagator depends on boundary functions e
J and e
Y , which are boundary
he homogeneous solutions:
Free propagator
Controls zero mode mass

Mixed position–momentum propagator
27
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

Incorporating of self-energy
Fichet 1905.05779 Section 5: Dressed Propagator
28
as they do in 4D Minkowski space. Including these e↵ects corresponds to evaluati
ng 1/N2
e↵ect on the propagator of the strongly coupled dual theory; in our case this
2
/k.
cus on bulk self-energy corrections from a cubic self-interaction. Brane-localized se
only modify the boundary conditions and are thus unimportant for our purposes.
o the free propagator, the Green’s function equation for the dressed propagator satisfi
DX (X, X0
)
1
p
g
Z
dY ⇧(X, Y ) (Y, X0
) =
i
p
g
(5)
(X X0
) , (4
calculation of diagrams in AdS has recently been an intense topic of research, see e.g. [51–56] for lo
ams and [57–60] for developments in position–momentum space. Throughout this paper we instead
te propagators.
9
n of motion, we may integrate the kinetic term by parts.
p
g|@ |2
= @M
p
g ⇤
gMN
@N
⇤
@M
p
ggMN
@N . (3.3
motion is
O = @M
p
ggMN
@N
p
gM2
= 0 . (3.4
operator maps onto:
O =
✓
R
z
◆3
"
@2
z
3
z
@z + p2
✓
R
z
◆2
M2
#
. (3.5
rts with respect to @z generates surface terms at R0
and R:
propagator + Π
eral case works similarly. The final result is found to be
Im⇧p(z, z0
) ⇡
1
8⇡
2 k2
2( + 1) 2( + 2)
1
(kz)3(kz0)3
✓
z<
4z>
◆2 +2
. (37)
At that point we have a simple expression for the 1PI insertion. However it is still
local and it is thus difficult to solve the dressed equation of motion. To go further
shall use a position space version of the narrow width approximation (NWA). The
ition space NWA amounts to a @5 expansion of ⇧ where the @5 derivatives act on the
pagator. 9 It can be equivalently seen as an expansion over the basis of the Dirac
a’s derivatives,
⇧p(z, z0
) = F0(z) (z z0
) F1(z) 0
(z z0
) +
F2(z)
2
00
(z z0
) + . . . . (38)
s can be directly obtained from the dressed equation of motion Eq. (31), where ⇧p
onvoluted with p. The coefficients of the expansion of the ⇧p distribution in Dirac
ta’s derivative are found to be given by the moments of the distribution, 10
Z
Local expansion of self-energy (5D narrow width approx.)
Affects argument of
homogeneous solutions
Small deformations of free solution

Incorporating self-energy
Fichet 1905.05779 ,Section 5: Dressed Propagator; quantitative results: Fichet & Costantino (in preparation)
29
2 5
and gives solutions where, in top of a complex Bessel function order and z-dependent
phase, the argument of the Bessel function is also changed. Showing only this last e↵ect
for simplicity, the solutions take the form
z2
J↵
pz
p
1 + iC2
2/2k
!
, z2
Y↵
pz
p
1 + iC2
2/2k
!
. (45)
This deformation is the important one, because it changes the phase of the Bessel’s
function argument. As a consequence, there are no poles along the real axis, and the
Bessel functions rather have an exponential behaviour controlled by the imaginary part
of the argument. The full propagator in the presence of this deformation is given by the
free propagator Eq. (11) where p is replaced by p/
p
1 + iC2
2/2k ⇡ p(1 iC2
2/4k).
In the p & 1/z> region, for timelike momentum the propagator behaves therefore as
p z, z0
/ e C2
2/4k pz>
. (46)
From this result we conclude that bulk interactions induce an exponential suppression
of the propagator: the IR region of the Lorentzian AdS background is opaque. This
is a quantum e↵ect, unlike the case of spacelike momentum where suppression occurs
in the free propagator, the suppression is here controlled by the interaction-dependent,
loop-induced parameter C2
2/4k.
The e
Y and e
J are shorthand for the boundary operator
xample:
BUV
⇥
z2
J↵(pz)
⇤
z=R
. (3.10)
ditions:
BIR
(z) = ↵IR
0
(R0
) + IR (R0
) = 0 . (3.11)
e boundaries. This, in turn, depends on the Lagrangian
1
p
R
X
n
f(n)
(z) (n)
(x) . (3.12)
✓
R
◆3 ✓
R
◆5
2
#
f(n)
(z)
Bulk self-interaction shifts
momentum into complex plane.
Higher KK modes have larger
widths (expected)
Re
Im

Opacity for timelike momenta
4D Narrow width approximation breaks down (due to interactions)
Higher Kaluza–Klein modes have more phase space for decays and hence larger
widths. Eventually these widths merge into a continuum rather than a sequence
of Breit–Wigner resonances. n.b. resonances are also not diagonal
See also Dynamical Dark Matter realization, Brooks, Dienes et al. 1610.04112, 1912.10588
30
Not to be confused with
Multiparticle continuum
one-“particle” states

ENERGY
SCA
E
What happens at Λμ/k ?
31
5D warped
Continuum states
Discrete KK modes
IR brane “decouples”
Similarity to unparticle dynamics is not coincidental, e.g. Friedland et al. 0902.3676

What happens to the ‘soft bomb’ event?
Produce a high-p mode on the UV brane…
More on soft bomb pheno, see Knapen et al. 1612.00850
32

Cascade Recursion Relation
See also, e.g. treatment of unstable particles in (review): Phys.Part.Nucl. 45 (2014), Kuksa “Unstable States in QFT”
33
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

Trick: replace KK sum with discontinuity across spectral representation
34
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

35
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

Cascade exponential suppression
Opacity at work
36
ENERGY
SCA
E
5D warped continuum
Continuum states
5D warped KK regime
Discrete KK modes

Killer Application
Annihilation into Mediators
• Not too heavy: mediators are in the KK
regime and are valid asymptotic states

• Heavy DM: mediators in the
continuum, rate appears to be
suppressed because narrow width
approximation breaks down
37
BULK
MEDI
U
B
E
I
B
E
Consider: dark matter on UV brane
annihilating into mediators
(e.g. for thermal freeze out)

Outline
Fractional-power self-interactions
38

Self-Interacting Dark Matter
Ignore visible matter: what is the signature of dark interactions?
Modern incarnation: Tulin, Yu, Zurek (1302.3898), Review: Tulin & Yu (1705.02358)
39
e 1: t-channel diagrams generating long-range forces in the
sent strong dynamics.
Mediator exchange induces a long range potential between dark matter. 
Dark matter scattering can exchange energy, halo cores become isothermal.
4D: Yukawa

2
s
s
-
-
y.
)
-
d
e
o
r
1 cm
2 êg
10 cm
2 êg
100 cm
2 êg
0.1 cm
2 êg
sêm = 0.01 cm
2 êg
10 50 100 500 1000 5000
1
10
102
103
104
Xv HkmêsL
Xsvêm
Hcm
2
êg
â
kmêsL
Self-Interacting Dark Matter
Kaplinghat, Tulin, Yu (1508.03339)
40
0.1 cm2/g
clusters
1 cm2/g
Dwarfs, LSB
simulation
mχ ~100 GeV
mΦ ~10 MeV
constant σv

SIDM: rich phase structure
Tulin, Yu, Zurek (1302.3898)
41
FIG. 1: Colored regions show parameter points (a, b) within our numerical scan, with the corresponding
values of T k2/(4⇡) (left) and `max (right) at each point. The classical, Born, and resonant regimes are
delineated by solid lines.
mX = 200 GeV
mf = 1 MeV
aX = 10-2
v = 1000 kmês
sT
clas
êmX
0 200 400 600 800 1000 1200 1400
-0.03
-0.02
-0.01
0.00
0.01
{max
s
T
êm
X
Hcm
2
êgL
mX=200 GeV
aX=10-2
v=10 kmês
Classical Resonant Born
0.001 0.01 0.1 1
10-8
10-5
0.01
10
104
107
mf HGeVL
s
T
êm
X
Hcm
2
êgL
FIG. 2: Left: Numerical calculation of T /mX, truncated at fixed `max, showing convergence with in-
creasing `max. The parameter point chosen corresponds to the classical regime with an attractive potential.
The convergence to the classical analytic result shown by dashed line. Right: Numerical calculation (solid
blue) of T /mX versus m , showing convergence to the classical analytical formula (dotted pink) and Born
Perturbative
(Born approx)
Non-perurbative
Coulomb scattering
Resonant
No analytic expressions
Fix velocity & coupling scan over mediator mass

What happens with an ensemble of mediators?
Sum of Yukawas with different ranges
Chaﬀey, Fichet, Tanedo (in progress)
42
= 1 + 2 + 3 + . . .
Figure 1: The t-channel exchange diagram can be represented as an infinite sum of t-channel exchanges
of KK modes. In this representation it is easy to see that the potential is an infinite sum of Yukawa
potentials. The existence of a zero mode depends on the values of ↵ and bUV.
2.4 Bulk Self-Interactions and the Continuum Regime
In (2.5) we neglected to consider higher order self-interactions for the bulk scalar. However in
principle, these interactions can exist and may lead to significant phenomenological e↵ects. Cubic
5D
X(z, z0
) =
N
X
n=n0
fn(z)fn(z0
)A(m2
n; z, z0
) . (10.2)
an convert the sum over kk wavefunctions into a contour integral over the magnitude of the 4-
ntum:
X(z, z0
) =
1
2⇡
I
C
d⇢ Gp
⇢(z, z0
)A(⇢; z, z0
) . (10.3)
ontour C is a series of counter-clockwise loops around the n0 through Nth poles. This is manifestly
with the kk decomposition of G(z, z0), but the real power is that one may replace it with the 5D
on-space propagator. Furthermore, one may deform the series of loops into a single counter-clockwise
ur that encloses the desired poles:
=) (10.4)
29
Sum over KK poles Discontinuity across branch cut

Fractional-power self-interacting potential
With Ian Chaﬀey and Sylvain Fichet
43
egime if one considers bulk self-interactions. Using the identities
nd (2z) = ⇡ 1/2
22z 1
(z) (z + 1/2) we find that the potential is gi
V (r) =
2
2⇡3/2
(3/2 ↵)
(1 ↵)
1
r
✓
1
kr
◆2 2↵
Q(2 2↵, m1r)
uppression on the scattering potential, standard numerical techniques fail since p
ymptotic states.
7
X(z, z0
) =
N
X
n=n0
fn(z)fn(z0
)A(m2
n; z, z0
) . (10.2
can convert the sum over kk wavefunctions into a contour integral over the magnitude of the 4
mentum:
X(z, z0
) =
1
2⇡
I
C
d⇢ Gp
⇢(z, z0
)A(⇢; z, z0
) . (10.3
e contour C is a series of counter-clockwise loops around the n0 through Nth poles. This is manifestl
e with the kk decomposition of G(z, z0), but the real power is that one may replace it with the 5D
ition-space propagator. Furthermore, one may deform the series of loops into a single counter-clockwis
tour that encloses the desired poles:
=) (10.4
29
Sum over KK poles Discontinuity across branch cut
IR regulator for mass gap
curvature
prefactor
Fractional power 0.5 < α < 1
Related to mediator bulk mass
There may also be brane-localized terms. Here M2
is a bulk mass parameter that will control t
profile of the zero mode. This is a particularly important parameter that shows up in the followi
combination
↵2
⌘ M2
R2
+ 4 . (3
A bulk scalar has mass dimension [ ] = 3/2. The BF bound states that ↵2
0. See (4.7)
Raman’s lectures [2]3
. The essence of this bound is that one may have negative bulk mass2
wh
perturbing about a stable vacuum because AdS provides a positive contribution to the energy.
3.1 Bulk Equation of Motion
To derive the equation of motion, we may integrate the kinetic term by parts.
p p p

Does it work?
Reproduces qualitative behavior of one-mediator SIDM
Chaﬀey, Fichet, Tanedo (in progress); Tulin, Yu, Zurek (1302.3898)
44
mX = 200 GeV
mf = 1 MeV
aX = 10-2
v = 1000 kmês
sT
clas
êmX
0 200 400 600 800 1000 1200 1400
-0.03
-0.02
-0.01
0.00
0.01
{max
s
T
êm
X
Hcm
2
êgL
mX=200 GeV
aX=10-2
v=10 kmês
Classical Resonant Born
0.001 0.01 0.1 1
10-8
10-5
0.01
10
104
107
mf HGeVL
s
T
êm
X
Hcm
2
êgL
IG. 2: Left: Numerical calculation of T /mX, truncated at fixed `max, showing convergence with in-
easing `max. The parameter point chosen corresponds to the classical regime with an attractive potential.
he convergence to the classical analytic result shown by dashed line. Right: Numerical calculation (solid

Behavior near resonances
Scaling becomes non-monotonic
45
Figure 4: (Left): Velocity dependence of the transfer cross section for a range of ↵. (Right): ↵ dependence
of the transfer cross section. For the top plots =
p
4⇡/10, m = 10 GeV, µ = 1 MeV, and k = 1000 TeV.
For the bottom plots =
p
4⇡. This reflects the qualitative behavior of a single mediator, see Figure 3
of Ref. [9].
3.1 Regulated Potential
We can improve (3.5) by instead evaluating the KK sum in (3.4). Using the integ
the potential can be written
V (r) =
2
8⇡2k
Z 1
m2
n0
d⇢ Disc⇢
⇥
p
⇢(R, R)
⇤ e
p
⇢r
r
where we have let m2
e
n ! 1. The discontinuity across the real axis is given by
Disc⇢
⇥
p
⇢(R, R)
⇤
=
1
k
✓
4k2
⇢
◆↵
(↵)
(1 ↵)
sin(⇡↵)
where we have assumed S↵ ⇡ ( 1)↵
. This validity of this approximation is im
continuum regime if one considers bulk self-interactions. Using the identities
⇡/ sin (⇡z) and (2z) = ⇡ 1/2
22z 1
(z) (z + 1/2) we find that the potential is giv
V (r) =
2
2⇡3/2
(3/2 ↵)
(1 ↵)
1
r
✓
1
kr
◆2 2↵
Q(2 2↵, m1r)
3
Without any suppression on the scattering potential, standard numerical techniques fail since p

Easy to fit…
More parameters
46
0.1 cm
2 /g
0.01 cm
2 /g
0.001 cm
2 /g
1 cm
2 /g
10 cm
2 /g
100 cm
2 /g
LINES OF CONSTANT
/m
[ / ]
/
[
/
×
/
]
Dwarfs
LSB
Clusters
Figure 5: Velocity dependence of the thermally averaged cross s
Yukawa, Yukawa = 7 × 10-4
, mX = 3 GeV, m = 8 MeV
= 0.955, X = 7, mX = 15 GeV, = 10.3 MeV
= 0.95, X = 0.036, mX = 1 GeV, = 411 keV
= 0.995, X = 0.03, mX = 100 MeV, = 59.3 keV
= 0.85, X = 0.011, mX = 100 MeV, = 28.94 keV
2
1 cm
2 êg
10 cm
2 êg
100 cm
2 êg
0.1 cm
2 êg
sêm = 0.01 cm
2 êg
10 50 100 500 1000 5000
1
10
102
103
104
Xv HkmêsL
Xsvêm
Hcm
2
êg
â
kmêsL
FIG. 1: Self-interaction cross section measured from astrophysical
data, given as the velocity-weighted cross section per unit mass as
a function of mean collision velocity. Data includes dwarfs (red),
LSBs (blue) and clusters (green), as well as halos from SIDM
N-body simulations with /m = 1 cm2
/g (gray). Diagonal
lines are contours of constant /m and the dashed curve is the
velocity-dependent cross section from our best-fit dark photon model
(Sec. V).
halo masses spanning 109
1015
M . These objects ex-
hibit central density profiles that are systematically shallower
than ⇢ / r 1
predicted from CDM simulations. To determine
the DM profile for each system, we perform a Markov Chain
Monte Carlo (MCMC) scan over the parameters (⇢0, 0, r1)
characterizing the SIDM halo, as well as the mass-to-light ra-
tio ⌥⇤ for the stellar density. The value for ⇢(r1) determines
the velocity-weighted cross section h vi/m from Eq. (1), as a
function of average collision velocity hvi = (4/
p
⇡) 0 for
a Maxwellian distribution. We also verify our model and
Tulin, Yu, Zurek (1302.3898)

What’s qualitatively new
SIDM + thermal relic?
For ordinary SIDM, annihilation rate is too large for thermal freeze out
typically assume asymmetric DM
For warped dark sector, that’s not the whole story. Production of large mediator KK
modes is exponentially suppressed due to breakdown of narrow width
approximation. Controlled by bulk self-coupling, not DM–mediator coupling.
“Have your cake and eat it too”
47
ent the thermally averaged cross section parametrically as h vi =
nd x = m /T. From (5.2) we can see that n = 1 and definin
m) we have
h vi0 =
e
n
X
n
e
n
X
m
f2
n(R)f2
m(R) e
A(mn, mm)⇥ (2m mn mm) .
a thermal relic, the cross section must satisfy
⌦ h2
= 3.51 ⇥ 10 9
GeV 2
p
g⇤(xf )xf
g⇤s(xf )h vi
 0.12
⇤s are evaluated at the freeze out temperature and
Expected annihilation rate:
Kinematically accessible
Final state profiles
(on UV brane)

Outline
48
Review: dark sectors
Review: 5D
opacity vs.
narrow widths
cascade decays
self-interactions
A few closing thoughts

Lots to explore
Very much a work in progress
• Consistency with AdS/CFT 
see upcoming work by Lexi Costantino
and Sylvain Fichet

• 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
• Dark photon 
Work with Kuntal Pal; see also recent
work by Rizzo et al. (e.g. 1801.08525)
Brax, Fichet, Tanedo (1906.02199) 49
FIMP? See also
Bernal et al.
2004.14403

Warped Dark Sector

Recommended

Recommended

More Related Content

Similar to Warped Dark Sector

Similar to Warped Dark Sector (20)

More from Flip Tanedo

More from Flip Tanedo (20)

Recently uploaded

Recently uploaded (20)

Warped Dark Sector