Vector Self-Interacting Dark Matter

f l i p . t a n e d o @ u c r . e d u 27
TEXAS A&M / MITCHELL INSTITUTE
Flip Tanedo
17 MAY 2019
UC Riverside Particle Theory
A SPIN-1 DARK SECTOR
& W H Y W E M I G H T C A R E
Work in progress with
Ian Chaffey (UCR)
f l i p . t a n e d o @ u c r . e d u TEXAS A&M / MITCHELL INSTITUTE
&

2
The challenge
I don’t think there’s a model of
vector dark matter with a low-mass
vector mediator.
e.g. kinetic mixing portal to vector dark matter
“ “”

3
This talk (and what it is not)
Build a dark sector with
vector dark matter, 
light vector mediator.
MASS
2
SU(2) GAUGE PION HIGGS
New model
New phenomena? 
(not this study)
Cool plots
New directions
Technical naturalness

4
Is this actually new?
KK VDM: Servant & Tait (hep-ph/0206071)
SU(2) VDM: Hambye (0811.0172), Gross et al. (1505.07480)
VDM + massless mediator: Baek et al. (1311.1035)
Topology: Murayama & Shu (0905.1720 ), Ko & Tang
(1609.02307), Khoze & Ro (1406.2291)
Simplified Model: Dent et al. (1505.03117)
Confined: Boddy et al. 1408.6532 & 1402.3629
Recent: Elahi & Khatibi 1902.04384, Choi et al. 1904.04109
Apologies for papers that I’ve missed… e-mail me soon if I missed yours.
This work: massive spin-1 mediator, no fermions.

5
Vector Dark Matter
SU(2)V
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g
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Wa
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COUPLING TRIPLET
We use common Standard Model particle names to 
emphasize analogy to SM symmetries.  
This talk: completely in the dark sector.

6
what we want
MASS
2
SU(2) GAUGE PION
Wa
what we’ve got
Model Building

7
Baek, Ko, Park (1311.1035 )
Model Building
TRIPLET
We parameterize the Goldstone fields
the broken generators [31]:
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generator
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
The gauged su(2)V symmetry is the dia
the covariant derivatives on the scalar
DµH = @µH igWa
µ Ta
H
h i =
1
2
✓
f
f
◆
=
?
SU(2) GAUGE
SU(2) GAUGE
h i =
2 f
=
ries su(2)H ! ? and su(2) ! u(1),
s
ds as spacetime-dependent transforma
'H · T =
p
2'+
HT+
+
p
2'H
p p
SU(2) GAUGE PION
want
Introduce triplet, give it a vev.
Breaks SU(2), gives mass to W±
stable

8
Baek, Ko, Park (1311.1035 )
Model Building
TRIPLET
SU(2) GAUGE
SU(2) GAUGE
SU(2) GAUGE PION
want
need to make this massive
h i =
2 f
=
ries su(2)H ! ? and su(2) ! u(1),
s
ds as spacetime-dependent transforma
'H · T =
p
2'+
HT+
+
p
2'H
p p

9
Baek, Ko, Park (1311.1035 )
Model Building
SU(2) GAUGE PION
want
hu
hd
=
1
2
3
2 +
p
2 3
he vacuum expectation values of the
hHi =
✓
0
v/
p
2
◆
h i
the global symmetries su(2)H ! ? an
be Goldstones
Figure 1: Spectrum. [Flip: Check this... I think
3 Symmetry Breaking
A linear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2
p
2
We parameterize the vacuum expectation values of th
hHi =
✓
0
v/
p
2
◆
h
These vevs break the global symmetries su(2)H ! ? a

10
Two problems
1. W± stability?
2. extra Goldstones?

11
Standard Model Interlude
SU(2)L ⇥ U(1)Y ! U(1)EM
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SU(2)L ⇥ SU(2)R ! SU(2)V
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GAUGE HIGGS
PIONS
W±
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A
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Z
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⇡±,0
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h
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'±
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'0
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GLOBAL SYMMETRY FOR EW
SU(2)H ⇥ SU(2)0
L ⇥ U(1)H
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EW is a gauged subgroup

12
Global Symmetry
al representation.
no interactions, the particles re
su(2) ⇥ su(2)H ⇥ u(1)H =
calar fields transform as
! U U†
su(2)H : H !
gonal (vector) subgroup su(2)V
(x) ! U (x)U†
,
unitary matrix and Ta
= 1
2
a
are the generato
articles respect. a global “flavor” symmetry
u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
as
2)H : H ! UHH u(1)H : H ! ei✓H
H .
p su(2)V of su(2) ⇥ su(2)H composed of transf
exp(i✓a
Ta
) is a 2 ⇥ 2 special unitary matrix an
mental representation.
mit of no interactions, the particles respect. a g
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥
the scalar fields transform as
) : ! U U†
su(2)H : H ! UHH
he diagonal (vector) subgroup su(2)V of su(2)
UH. The orthogonal combination is the ‘axial’
Higgs number” symmetry is analogous to hype
! UH(x) (x) ! U (x)U
2 ⇥ 2 special unitary matrix and Ta
= 1
2
a
are
ntation.
ctions, the particles respect. a global “flavor” s
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H
s transform as
†
su(2)H : H ! UHH u(1)H : H
tor) subgroup su(2)V of su(2) ⇥ su(2)H compo
{
gauge diagonal subgroup: UΦ = UH
TRIPLET (REAL)
DOUBLET

13
Global Symmetry Breaking
I think the h is actually lifted to m2
h ⇠ µf.]
p
2 +
3
◆
±
⌘
1
+ i 2
p
2
. (3.1)
of the fields by
h i =
1
2
✓
f
f
◆
= fT3
. (3.2)
! ? and su(2) ! u(1), respectively.
me-dependent transformations of the vacuum by
Figure 1: Spectrum. [Flip: Check this... I
3 Symmetry Breaking
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2
We parameterize the vacuum expectation values o
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetries su(2)H !
3.1 Would-be Goldstones
al representation.
no interactions, the particles re
su(2) ⇥ su(2)H ⇥ u(1)H =
calar fields transform as
! U U†
su(2)H : H !
gonal (vector) subgroup su(2)V
{
gauge diagonal subgroup

14
Leftover symmetry
) is a 2 ⇥ 2 special unitary matrix and T = 2
are t
representation.
o interactions, the particles respect. a global “flavor” sy
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
lar fields transform as
! U U†
su(2)H : H ! UHH u(1)H : H !
nal (vector) subgroup su(2)V of su(2) ⇥ su(2)H compos
orthogonal combination is the ‘axial’ symmetry su(2)A
umber” symmetry is analogous to hypercharge in the St
Renormalizable Lagrangian
alizable Lagrangian satisfying the global symmetries of
= exp(i✓ T ) is a 2 ⇥ 2 special unitary matrix and T =
damental representation.
limit of no interactions, the particles respect. a global “
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A
ch the scalar fields transform as
(2) : ! U U†
su(2)H : H ! UHH u
the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)
UH. The orthogonal combination is the ‘axial’ symme
“Higgs number” symmetry is analogous to hypercharge
eneral, Renormalizable Lagrangian
al, renormalizable Lagrangian satisfying the global symm
⇥ 2 special unitary matrix and T = 2
are the genera
ation.
ons, the particles respect. a global “flavor” symmetry
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
transform as
su(2)H : H ! UHH u(1)H : H ! ei✓H
H .
r) subgroup su(2)V of su(2) ⇥ su(2)H composed of tran
nal combination is the ‘axial’ symmetry su(2)A, for whic
ymmetry is analogous to hypercharge in the Standard M
malizable Lagrangian
agrangian satisfying the global symmetries of the partic
T ) is a 2 ⇥ 2 special unitary matrix and T = 2
are
representation.
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H
alar fields transform as
! U U†
su(2)H : H ! UHH u(1)H : H !
onal (vector) subgroup su(2)V of su(2) ⇥ su(2)H compo
e orthogonal combination is the ‘axial’ symmetry su(2)
umber” symmetry is analogous to hypercharge in the S
malizable Lagrangian satisfying the global symmetries of
do not include any such terms.
2 Spectrum, Symmetry, Stability
ualitative overview of the model is as follows. The vacuum o
aks the global symmetry su(2) ⇥ su(2)H ⇥ u(1)H ! u(1)H0
u(1)H0 : T3
V +
1
2
TH ,
logous to electric charge in the electroweak sector. In wha
ark sector particle with respect to the u(1)V ⇢ su(2)V gaug
ge bosons eat three of the five Goldstone modes. We sug
ns,’ ⇡±
. We take the limit where the triplet vev is much la
hTr 2
i =
f2
2
h|H|2
i =
we do not include any such terms.
2.2 Spectrum, Symmetry, Stability
A qualitative overview of the model is as follows. The vacuum
breaks the global symmetry su(2) ⇥ su(2)H ⇥ u(1)H ! u(1
u(1)H0 : T3
V +
1
2
TH
analogous to electric charge in the electroweak sector. In w
a dark sector particle with respect to the u(1)V ⇢ su(2)V g
gauge bosons eat three of the five Goldstone modes. We
pions,’ ⇡±
. We take the limit where the triplet vev is much
hTr 2
i =
f2
2
h|H|
{
Figure 1: Spectrum. [Flip: Check this
3 Symmetry Breaking
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2
We parameterize the vacuum expectation value
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetries su(2)H
+1/2
-1/2
+1
+1
TV3 TH
Analog of electromagnetism
after electroweak breaking
gauged
U(1)H’ stabilizes dark matter

15
Leftover symmetry
are t
representation.
! U U†
su(2)H : H ! UHH u(1)H : H !
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A
(2) : ! U U†
su(2)H : H ! UHH u
are the genera
ation.
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
transform as
su(2)H : H ! UHH u(1)H : H ! ei✓H
H .
are
representation.
! U U†
su(2)H : H ! UHH u(1)H : H !
do not include any such terms.
2 Spectrum, Symmetry, Stability
ualitative overview of the model is as follows. The vacuum o
aks the global symmetry su(2) ⇥ su(2)H ⇥ u(1)H ! u(1)H0
u(1)H0 : T3
V +
1
2
TH ,
logous to electric charge in the electroweak sector. In wha
ark sector particle with respect to the u(1)V ⇢ su(2)V gaug
ge bosons eat three of the five Goldstone modes. We sug
ns,’ ⇡±
. We take the limit where the triplet vev is much la
hTr 2
i =
f2
2
h|H|2
i =
we do not include any such terms.
A qualitative overview of the model is as follows. The vacuum
breaks the global symmetry su(2) ⇥ su(2)H ⇥ u(1)H ! u(1
u(1)H0 : T3
V +
1
2
TH
analogous to electric charge in the electroweak sector. In w
a dark sector particle with respect to the u(1)V ⇢ su(2)V g
gauge bosons eat three of the five Goldstone modes. We
pions,’ ⇡±
. We take the limit where the triplet vev is much
hTr 2
i =
f2
2
h|H|
{
Figure 1: Spectrum. [Flip: Check this
3 Symmetry Breaking
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2
We parameterize the vacuum expectation value
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetries su(2)H
+1/2
-1/2
+1
+1
TV3 TH
gauged
order parameter of U(1) breaking
is charge 1/2
incompatible
with SU(2)V

16
Two problems
1. W± stability?
2. extra Goldstones?

17
Goldstone smorgasborg
More Goldstones
than gauge bosons

Leftovers are pions.

18
Leftover symmetry
are t
representation.
! U U†
su(2)H : H ! UHH u(1)H : H !
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A
(2) : ! U U†
su(2)H : H ! UHH u
are the genera
ation.
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
transform as
su(2)H : H ! UHH u(1)H : H ! ei✓H
H .
are
representation.
! U U†
su(2)H : H ! UHH u(1)H : H !
A qualitative overview of the model is as follows. T
breaks the global symmetry su(2) ⇥ su(2)H ⇥ u(
u(1)H0 :
analogous to electric charge in the electroweak se
a dark sector particle with respect to the u(1)V ⇢
gauge bosons eat three of the five Goldstone mo
‘pions,’ ⇡±
. We take the limit where the triplet v
hTr 2
i =
f2
2
{
gauged
EATEN GOLDSTONES
UNEATEN PIONS
Explicitly break SU(2)A,  
give mass to pions
PSEUDO-GOLDSTONES
)V ⇥ su(2)A ⇥ u(1)H , (2.2)
H u(1)H : H ! ei✓H
H . (2.3)
(2) ⇥ su(2)H composed of transformations
ial’ symmetry su(2)A, for which U = U†
H.
hypercharge in the Standard Model.
an
global symmetries of the particle content is
V (2.4)
2
+ µH†
H + 00
|H|2
Tr 2
. (2.5)
al and adjoint of su(2), respectively. We
H obtain vacuum expectation values (vevs)
ry. This breaking produces a spectrum of
EXPLICIT SU(2)A
BREAKING

19

20
The Model
e the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)H composed of trans
= UH. The orthogonal combination is the ‘axial’ symmetry su(2)A, for which
H “Higgs number” symmetry is analogous to hypercharge in the Standard Mo
General, Renormalizable Lagrangian
eral, renormalizable Lagrangian satisfying the global symmetries of the particle
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ |2
V
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
.
D are covariant derivatives for the fundamental and adjoint of su(2), respect
e potential V to imply that the scalars and H obtain vacuum expectation va
ntaneously break the symmetries of the theory. This breaking produces a sp
ne bosons, three of which are eaten by the massive gauge bosons. The triline
y breaks the global axial su(2)A symmetry. This gives a mass to the remaining
stone modes. The 00
term mixes the radial modes of the H and . We syst
the theory starting from the symmetry breaking and 0
terms and subsequen
ts of the µ and 00
terms. Additional quartic terms obeying the global symmet
0 1
{
{
Figure 1: Spectrum. [Flip: Check this... I think the h is actually lifted to m2
h ⇠ µf.]
Symmetry Breaking
ear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
±
⌘
1
+ i 2
p
2
.
arameterize the vacuum expectation values of the fields by
hHi =
✓
0
v/
p
2
◆
h i =
1
2
✓
f
f
◆
= fT3
.
e vevs break the global symmetries su(2)H ! ? and su(2) ! u(1), respectively.
Would-be Goldstones
arameterize the Goldstone fields as spacetime-dependent transformations of the
roken generators [31]:
H = ei
'H ·T
v/2 hHi 'H · T =
p
2'+
HT+
+
p
2'HT + '0
HT3
his... I think the h is actually lifted to m2
h ⇠ µf.]
is
3
p
2 +
2 3
◆
±
⌘
1
+ i 2
p
2
. (3.1)
ues of the fields by
h i =
1
2
✓
f
f
◆
= fT3
. (3.2)
)H ! ? and su(2) ! u(1), respectively.
cetime-dependent transformations of the vacuum by
' · T =
p
2'+
T+
+
p
2' T + '0
T3
(3.3)
MASS TO VECTORS
EXPLICIT SU(2)A
BREAKING
MASS TO PIONS
SPONTANEOUS BREAKING
H =
✓
hu
hd
◆
=
We parameterize the vacuum expectati
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetrie
We parameterize the Goldstone fields
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generators
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
3 Symmetry Breaking
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
We parameterize the vacuum expectation values of the fields
hHi =
✓
0
v/
p
2
◆
h i =
1
2
These vevs break the global symmetries su(2)H ! ? and su
We parameterize the Goldstone fields as spacetime-depende
H = ei
'H ·T
v/2 hHi 'H · T =
p
2'
= ei
' ·T
f h i e i
' ·T
f ' · T =
p
2'
with respect to the su(2)H, generators T±
= T1
± iT2
, T3
. T
✓ ◆
DOUBLET
TRIPLET
V H
= UH. The orthogonal combination is the ‘axial’ symmetry su(2)A, for which
H “Higgs number” symmetry is analogous to hypercharge in the Standard Mo
General, Renormalizable Lagrangian
eral, renormalizable Lagrangian satisfying the global symmetries of the particle
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ |2
V
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
.
D are covariant derivatives for the fundamental and adjoint of su(2), respect
e potential V to imply that the scalars and H obtain vacuum expectation va
ntaneously break the symmetries of the theory. This breaking produces a sp
ne bosons, three of which are eaten by the massive gauge bosons. The triline
y breaks the global axial su(2)A symmetry. This gives a mass to the remaining
stone modes. The 00
term mixes the radial modes of the H and . We syst
the theory starting from the symmetry breaking and 0
terms and subsequen
ts of the µ and 00
terms. Additional quartic terms obeying the global symmet
0
term.1
✓ ◆
Full disclosure: there’s a tuning in this model!

21
Portals to Standard Model
Higgs mixing: e.g. Baek et al. (1311.1035; dark Z: Lee et al. 1203.2947
|HSM|2
a|H2
| + bTr 2
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1
⇤
˜AB
WA
µ⌫WB µ⌫
SM
<latexit sha1_base64="0DxeESrkCWU/c09rKnRbuTWS+ZQ=">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</latexit>
1
⇤
A
WA
µ⌫Bµ⌫
SM
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Higgs mixing portal
Z-mixing
Kinetic mixing
THOROUGHLY EXPLORED

22
Monopoles?
Georgi—Glashow model;
has t’Hooft—Polyakov
monopoles.
SU(2) GAUGE TRIPLET
0
SU(2) GAUGE
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H
D and D are covariant derivatives for the fund
write the potential V to imply that the scalars
that spontaneously break the symmetries of the
Goldstone bosons, three of which are eaten by
explicitly breaks the global axial su(2)A symmetr
be Goldstone modes. The 00
term mixes the ra
examine the theory starting from the symmetry b
the e↵ects of the µ and 00
terms. Additional qua
to the 00
term.1
1
For example: H† 2
H =
1
2
H†
{ , } H =
1
2
H†
✓
1
2
a
3

23
Dark sector monopoles
don’t do much here.
Sequestered from visible
matter.
1311.1035
SU(2) GAUGE TRIPLET
0
SU(2) GAUGE
mA⟷ confinement 
Kinetic mixing
BUT IN OUR CASE:
MASSLESS

24
Kinetic Mixing with Monopoles
visible matter picks up small dark charge
dark stuff stays dark
Kinetic Mixing:

25
Kinetic Mixing with Monopoles
Anson Hook and Junwu Huang 1705.01107
visible matter picks up small dark charge
dark stuff stays dark
visible magnetic
charge
picks up small
dark magnetic
monopole
Kinetic Mixing:
Implies:
millimagnetic charge

26
1808.09459; FT, Can Kilic, Mario Martone in progress.
Tip of the iceberg

27
thanks!
Figure 8: Cartoon of the Goldstone excitation for a ‘Mexican hat’ potential. Image from [148].
4.3.1 Framework
We begin with the concrete example of low-energy qcd that we described above. Given that the
chiral condensate hq̄qi breaks SU(3)A, we proceed to write down the e↵ective theory describing
the interaction of the resulting Goldstone bosons. Let us write U0 to refer to the direction in field
space associated with the chiral condensate, U0 ⇠ hq̄qi. This transforms as a bifundamental with
respect to SU(3)L⇥SU(3)R,
U(x) ! ULU(x)U†
R, (4.9)
where UL and UR are the transformation matrices under the SU(3)L and SU(3)R respectively. The
observation that SU(3)A is broken corresponds to U0 = . Note that this indeed preserves the
SU(3)V transformations UL = UR.
We now consider the fluctuations U(x) about U0—these are what we identify with the Goldstone
bosons. Recall the picture of spontaneous symmetry breaking through the ‘Mexican hat’ potential
in Fig. 8. The action of an unbroken symmetry does not a↵ect the vev (represented by the ball),
while broken symmetries shift the vev along the vacuum manifold. This gives an intuitive picture
of how to identify the Goldstone modes:
1. Identify a convenient vev, U0

28
Additional Slides

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
29
Self-Interacting Dark Matter
Kaplinghat, Tulin, Yu 1508.03339
10 GeV DM
10 MeV med
DWARFS
LSB
CLUSTER
N-BODY

30
Goldstone space
on values fa > fb. The Goldstone excitations
2
Only hHi breaks the u(1) symmetry so tha
contrast to the charged states for which th
pair of charged gauge bosons; see Appendi
identify the normalized su(2)V Goldstone '
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
In unitary gauge, '±
V only appears as the lo
is an uneaten Goldstone boson that remain
them ⇡±
in anticipation of including explici
3.4 Symmetry Breaking with
The simplest form of this model takes only
V | , 0 =
4!
2 Tr
These terms separately break the su(2) an
he linear combination of Goldstone bosons associated with su(2)V . Gau
n su(2)V promotes this global symmetry to a local symmetry. In unit
local su(2)V transformation to remove 'V from the theory. It appear
l polarization of the massive gauge bosons. We may express 'V in ter
ng this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA .
breaks the u(1) symmetry so that the A eats the only neutral Goldsto
the charged states for which there are two pairs of charged Goldstone
rged gauge bosons; see Appendix A for an illustrative u(1) example. F
normalized su(2)V Goldstone 'V and the orthogonal state 'A:
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
'±
A =
f'±
H (v/2)'±
p
f2 + (v/2)2
.
gauge, '±
V only appears as the longitudinal mode of W±
. The ‘axial’ c
en Goldstone boson that remains in the theory. We refer to these as pio
anticipation of including explicit symmetry breaking terms to make th

31
e.g. Direct Detection
Dent et al. 1505.03117
TABLE X. Non-relativistic reduction of operators for a spin-1 WIMP
Scalar Mediator
X†
µXµq̄q ≠æ
3
b1hN
1
m2
„
4
O1
X†
µXµq̄“5q ≠æ
3
b1hN
2
m2
„
4
O10
Vector Mediator
(X†
‹ˆµX‹ ≠ ˆµX†
‹X‹)(q̄“µq) ≠æ 0
(X†
‹ˆµX‹ ≠ ˆµX†
‹X‹)(q̄“µ“5q) ≠æ
3
≠3b5hN
4
m2
G
mN
mX
4
O10
ˆ‹(X‹†Xµ + X†
µX‹)(q̄“µq) ≠æ
3
Re(b6)hN
3
m2
G
mN
mX
4
(O5 + O6 ≠ q2
m2
N
O4)
ˆ‹(X‹†Xµ + X†
µX‹)(q̄“µ“5q) ≠æ
3
≠
2Re(b6)hN
4
m2
G
mN
mX
4
O9
ˆ‹(X‹†Xµ ≠ X†
µX‹)(q̄“µq) ≠æ
3
≠
4Im(b6)hN
3
m2
G
mN
mX
4
O17
ˆ‹(X‹†Xµ ≠ X†
µX‹)(q̄“µ“5q) ≠æ
3
4Im(b6)hN
4
m2
G
mN
mX
4
O18
‘µ‹fl‡
1
X‹†ˆflX‡ + X‹ˆflX‡†
2
(q̄“µq) ≠æ
3
Re(b7)hN
3
m2
G
mN
mX
4
O11
‘µ‹fl‡
1
X‹†ˆflX‡ + X‹ˆflX‡†
2
(q̄“µ“5q) ≠æ
3
Re(b7)hN
4
m2
G
mN
mX
4
(i q2
mX mN
O4 ≠ imN
mX
O6 ≠ 2O14)
‘µ‹fl‡
1
X‹†ˆflX‡ ≠ X‹ˆflX‡†
2
(q̄“µq) ≠æ
3
2Im(b7)hN
3
m2
G
4
(O8 + O9)
‘µ‹fl‡
1
X‹†ˆflX‡ ≠ X‹ˆflX‡†
2
(q̄“µ“5q) ≠æ
3
4Im(b7)hN
4
m2
G
4
O4
Charged Spinor Mediator
(X†
µX‹)(q̄“µ“‹q) ≠æ
3
y†
3y3≠y†
4y4
mQmX
4 1
fN
TqO1 + 2”N
q O4
2
3 † †
4
SI from Higgs
(erratum soon)

32
Effect of mu
μ encodes post-NLΣM interactions of pions
-10 -5 0 5 10
-10
-5
0
5
10
-10 -5 0 5 10
-10
-5
0
5
10
-10 -5 0 5 10
-10
-5
0
5
10
f
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v
<latexit sha1_base64="LiPx1XUPb+UtFns1m3WmqKTBbTE=">AAAB6HicdVDLSsNAFL2pr1pfVZduBovgKiSm0LorunHZgn1AG8pkOmnHTiZhZlIooV/gxoUibv0kd/6N0xeo6IELh3Pu5d57goQzpR3n08ptbG5t7+R3C3v7B4dHxeOTlopTSWiTxDyWnQArypmgTc00p51EUhwFnLaD8e3cb0+oVCwW93qaUD/CQ8FCRrA2UmPSL5Yc27v2KmUPLYnnrkkVubazQAlWqPeLH71BTNKICk04VqrrOon2Myw1I5zOCr1U0QSTMR7SrqECR1T52eLQGbowygCFsTQlNFqo3ycyHCk1jQLTGWE9Ur+9ufiX1011WPUzJpJUU0GWi8KUIx2j+ddowCQlmk8NwUQycysiIywx0Sabgglh/Sn6n7SubNex3Ua5VLtZxZGHMziHS3ChAjW4gzo0gQCFR3iGF+vBerJerbdla85azZzCD1jvX2KEjVI=</latexit>
µ
<latexit sha1_base64="9g8EBdnnKp0xDPeBkQpAcaTZhyk=">AAAB6nicdVDLSsNAFL3xWeur6tLNYBFchaTWJsuiG5cV7QPaUCbTSTt0JgkzE6GEfoIbF4q49Yvc+TdOH4KKHrhwOOde7r0nTDlT2nE+rJXVtfWNzcJWcXtnd2+/dHDYUkkmCW2ShCeyE2JFOYtpUzPNaSeVFIuQ03Y4vpr57XsqFUviOz1JaSDwMGYRI1gb6bYnsn6p7NjuhetXasixq37N8z1DvKp7XvOQaztzlGGJRr/03hskJBM01oRjpbquk+ogx1Izwum02MsUTTEZ4yHtGhpjQVWQz0+dolOjDFCUSFOxRnP1+0SOhVITEZpOgfVI/fZm4l9eN9ORH+QsTjNNY7JYFGUc6QTN/kYDJinRfGIIJpKZWxEZYYmJNukUTQhfn6L/Satiuyarm2q5frmMowDHcAJn4IIHdbiGBjSBwBAe4AmeLW49Wi/W66J1xVrOHMEPWG+f4M2OMQ==</latexit>
metry Breaking with , 0
, µ
e µ term in the potential explicitly breaks su(2) ⇥ su(2)H ! su(2)V
oportional to µ:
V | , 0,µ =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H .
minimum of the potential from (3.13) to the following condition:
f2
= f2
0 +
3µv2
2 f
v2
= v2
0 +
3µf
0
.

33
Pion mediator?
Ann Nelson et al. 1809.09667; FT and Javi Serra in progress
Dark Matter Mediator
Standard
Model
Higgs
composite
DIRAC FERMION
SO(6)/SO(5)
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SO(6) SO(4) ⇠
= SU(2)L ⇥ SU(2)R
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GLOBAL SYMMETRY

34
An SO(6) basis
Reminder: SO(6) ⊃ SO(4) ≃ SU(2)L x SU(2)R (custodial symmetry)
MEDIATOR
U(1) SYM
BROKEN
⌃0 = (0, 0, 0, 0, 0, 1)T
VEV

35
Goldstone Parameterization
the generator basis given in Appendix A.1,
U =
1
f
0
B
B
B
B
@
f13⇥3 · · ·
· f h2
f+
p
f2 h2 ⌘2
h⌘
f+
p
f2 h2 ⌘2
h
· h⌘
f+
p
f2 h2 ⌘2
f ⌘2
f+
p
f2 h2 ⌘2
⌘
· h ⌘
p
f2 h2 ⌘2
1
C
C
C
C
A
he NGB decay constant, and the entries with a dot are zero. In this basi
metry acts on the upper-left 4⇥4 block. We project out the unbroken SO
eft 5 ⇥ 5 block, by contracting with the vector ⌃0 = (0 0 0 0 0 1)T
. Fro
eld
⌃ = U⌃0 =
1
f
(0 0 0 h ⌘
p
f2 h2 ⌘2)T
,
se to construct SO(6) invariants.
etic terms and non-linear interactions between the NGBs, to leading or
derivatives, O(@2
), is given by the invariant operator
2 ✓
U(1) associated with singlet
Goldstones: act on vacuum  
with broken generators, promote  
transformation parameters to fields
UNITARY GAUGE
(REMOVE EATEN GOLDSTONES)

36
Symmetry Structure

37
What the theory looks like
NLΣM
dark matter
embedding
visible matter
embedding
CHOICES LEAD TO 
- HIGGS POTENTIAL
- MEDIATOR POTENTIAL
CHOICES LEAD TO
MEDIATOR POTENTIAL
IRREDUCIBLE 
INTERACTIONS
OUTPUTS:
MEDIATOR MASS AS A 
FUNCTION OF DISCRETE 
CHOICES AND DM MASS
COMPOSITE HIGGS
SPECIFY BREAKING SCALE f 
GIVES MEASURE OF TUNING

Vector Self-Interacting Dark Matter

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