Vector Self-Interacting Dark Matter

f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
Flip Tanedo
1 MAY 2019
UC Riverside Particle Theory
SPIN-1 DARK MATTER
& W H Y W E M I G H T C A R E
Work in progress with
Ian Chaffey
f l i p . t a n e d o @ u c r . e d u TEXAS A&M / MITCHELL INSTITUTE
&

2
Team Flip 
March 2019
Ian Chaffey
Ian Chaffey
work with

3
the plan
via pexels.com
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
SM
med
DM

4
WIMP Complementarity
Dark matter searches related by crossing symmetry:
How Dark Matter talks to the Standard Model
.
.
χ
.
χ
.
sm
.
sm
.
.
χ
.
sm
.
χ
.
sm
.
.
sm
.
sm
.
χ
.
χ
A
N
N
I
H
I
L
ATI
ON
D
I
RECT DETECTIO
N
COLLI
DER
INDIRECT DIRECT COLLIDER
Standard Model
Dark Matter
A
N
N
I
H
I
L
ATI
ON
COLLI
DER
D I R E C T
WEAK FORCE
R E L I C A B U N DA N C E YO U ’ R E K I L L I N G M E N OT G R E AT, E I T H E R

5
Light Mediators
e
e
e
e
e
capture
a
n
n
i
h
i
l
a
t
i
o
n
x x
A0
A0
INDIRECT DIRECT COLLIDER
Standard Model
Mediator
N N
q
q
A
N
N
I
H
I
L
ATI
ON
COLLI
DER
D I R E C T
Dark Matter
can keep thermal relic!

6
New Searches with Light Mediators
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 MEDIATOR PRODUCTION
N N
A
N
N
I
H
I
L
ATI
ON
COLLI
DER
D I R E C T
A0
A0
Halo Morpholo
• SIDM particles follow the
0 2 4 6 8
0
2
4
6
8
R HkpcL
z
HkpcL
constant density contours
Kaplinghat, Linden, Keeley, HBY (2013) (P
C
d
SELF
Standard Model
Mediator
Dark Matter
SM
SM
SM
SM
accelerators astro
R E L I C  
A B U N DA N C E

7
This talk
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

8
Is this actually new?
KK DM: Servant & Tait (hep-ph/0206071)
SU(2) VDM: Hambye (0811.0172), Gross et al. (1505.07480)
Topology: Murayama & Shu (0905.1720 ), Baek et al. (1311.1035),
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
This work: massive spin-1 mediator, no fermions.

9
the plan
via pexels.com
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
SM
med
DM

10
the plan
via pexels.com
1. symmetry structure
2. model building
3. hint of pheno

11
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.  
Most of talk: completely in the dark sector.

12
Baek, Ko, Park (1311.1035 )
Vector Dark Matter
SU(2) GAUGE TRIPLET
These vevs break the global symmetrie
3.1 Would-be Goldstones
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, generators
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
h i =
1
2
✓
f
f
◆
=
?
0 massless dark photon
SU(2) GAUGE
h i =
2 f
=
ries su(2)H ! ? and su(2) ! u(1), r
s
ds as spacetime-dependent transforma
'H · T =
p
2'+
HT+
+
p
2'H
p p

13
Baek, Ko, Park (1311.1035 )
Vector Dark Matter
SU(2) GAUGE TRIPLET
0
SU(2) GAUGE
let’s make this massive
need to break U(1)
h i =
2 f
=
s
'H · T =
p
2'+
HT+
+
p
2'H
p p

14
Vector Dark Matter
Hi =
✓
0
v/
p
2
◆
h i
he global symmetries su(2)H ! ? and
e Goldstones
he Goldstone fields as spacetime-depe
ors [31]:
'H ·T
v/2 hHi ' · T =
p
h i =
2 f
=
s
'H · T =
p
2'+
HT+
+
p
2'H
p p
W stability? 
Extra particles?

15
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
SU(2)H ⇥ SU(2)0
L ⇥ U(1)H
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EW is a gauged subgroup

16
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
TRIPLET (REAL)
DOUBLET

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
A linear parameterization of the scalar fields is
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 !
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

18
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

19
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
◆
+1/2
-1/2
+1
+1
TV3 TH
gauged
order parameter of U(1) breaking
is charge 1/2

20
Goldstone smorgasborg
More Goldstones
than gauge bosons

Leftovers are pions.

+ radial modes
EAT
EAT

21
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

22
Standard Model Interlude
SU(2)L ⇥ U(1)Y ! U(1)EM
SU(2)L ⇥ SU(2)R ! SU(2)V
GAUGE HIGGS
PIONS
W±
A
Z
⇡±,0
h
'±
'0
GLOBAL SYMMETRY
SU(2)H ⇥ SU(2)0
L ⇥ U(1)H
Mass from explicit
breaking of global sym.
FROM QUARK MASSES

23
the plan
via pexels.com
1. symmetry structure
2. model building
3. hint of pheno

24
Potential H =
✓
hu
hd
◆
=
We parameterize the vacuum expectati
hHi =
✓
0
v/
p
2
◆
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
H|radial =
1
p
2
✓
0
h
◆
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
Figure 2: Fields a and b acquire unequal vacuum expectation values fa > fb. The Goldstone excitations
with respect to a transformation by parameter ✓ have correspondingly di↵erent magnitudes, 'a > 'b.
The Goldstone, 'V , for a vectorial transformation where ✓a = ✓b is thus not orthogonal to the corre-
sponding Goldstone, 'A for an axial transformation where ✓a = ✓b.
u(1)V and u(1)A by the same e↵ective order parameter, f2
V = f2
a + f2
b . Neither u(1)V nor u(1)A is
preferred over the other. Why, then, is it the case in (A.7) that the 'V eats more 'a while 'A
eats more of 'b? The root of this confusion is illustrated in Fig. 2: in the absence of gauging, the
naı̈ve description of the vector and axial Goldstones are not orthogonal to one another. The choice
of gauging a particular combination of the full global symmetry breaks the symmetry and gives
‘priority’ to the eaten Goldstone boson to have a larger admixture of the field that does most of
the symmetry breaking.
A.4 Gauging an Axial Combination
Figure 2: Fields a and b acquire unequal vacuum expectation values fa > fb. The Goldstone excitations
with respect to a transformation by parameter ✓ have correspondingly di↵erent magnitudes, 'a > 'b.
The Goldstone, 'V , for a vectorial transformation where ✓a = ✓b is thus not orthogonal to the corre-
sponding Goldstone, 'A for an axial transformation where ✓a = ✓b.
u(1)V and u(1)A by the same e↵ective order parameter, f2
V = f2
a + f2
b . Neither u(1)V nor u(1)A is
preferred over the other. Why, then, is it the case in (A.7) that the 'V eats more 'a while 'A
eats more of 'b? The root of this confusion is illustrated in Fig. 2: in the absence of gauging, the
naı̈ve description of the vector and axial Goldstones are not orthogonal to one another. The choice
of gauging a particular combination of the full global symmetry breaks the symmetry and gives
‘priority’ to the eaten Goldstone boson to have a larger admixture of the field that does most of
the symmetry breaking.
A.4 Gauging an Axial Combination
su(2) : ! U U su(2)H : H ! UHH u(1)H : H ! e H .
ge 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), respec
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 trilin
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
{
{
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 ·T p p
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
SPONTANEOUS BREAKING

25
The Model
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
.
stone modes. The 00
terms and subsequen
ts of the µ and 00
0 1
{
{
h ⇠ µf.]
Symmetry Breaking
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
±
⌘
1
+ i 2
p
2
.
hHi =
✓
0
v/
p
2
◆
h i =
1
2
✓
f
f
◆
= fT3
.
Would-be Goldstones
H = ei
'H ·T
v/2 hHi 'H · T =
p
2'+
HT+
+
p
2'HT + '0
HT3
h ⇠ µf.]
is
3
p
2 +
2 3
◆
±
⌘
1
+ i 2
p
2
. (3.1)
h i =
1
2
✓
f
f
◆
= fT3
. (3.2)
' · T =
p
2'+
T+
+
p
2' T + '0
T3
(3.3)
EAT GOLDSTONES
(MASS TO VECTORS)
EXPLICIT SU(2)A
BREAKING
(MASS TO SCALARS)
H =
✓
hu
hd
◆
=
We parameterize the vacuum expectati
hHi =
✓
0
v/
p
2
◆
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
H|radial =
1
p
2
✓
0
h
◆
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'
= T1
± iT2
, T3
. T
✓ ◆
DOUBLET
TRIPLET
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
.
stone modes. The 00
terms and subsequen
ts of the µ and 00
0
term.1
✓ ◆

26
Gauge Boson Mass
EAT GOLDSTONES
(MASS TO VECTORS)
DOUBLET
TRIPLET
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
.
stone modes. The 00
terms and subsequen
ts of the µ and 00
0
term.1
✓ ◆
hHi =
✓
0
v/
p
2
◆
We parameterize the Goldstone fields as space
H = ei
'H ·T
v/2 hHi '
= ei
' ·T
f h i e i
' ·T
f '
= T
H|radial =
1
p
2
✓
0
h
◆
The gauged su(2)V symmetry is the diagonal com
the covariant derivatives on the scalar fields ar
DµH = @µH igWa
µ Ta
H
H =
hd
=
2 2 3 ⌘ p
We parameterize the vacuum expectation values of the fields by
hHi =
✓
0
v/
p
2
◆
h i =
1
2
✓
f
f
◆
= fT3
.
These vevs break the global symmetries su(2)H ! ? and su(2) ! u(1), respecti
We parameterize the Goldstone fields as spacetime-dependent transformations o
H = ei
'H ·T
v/2 hHi 'H · T =
p
2'+
HT+
+
p
2'HT + '
= ei
' ·T
f h i e i
' ·T
f ' · T =
p
2'+
T+
+
p
2' T ,
= T1
± iT2
, T3
. The radial modes are
H|radial =
1
p
2
✓
0
h
◆
|radial =
1
2
✓ ◆
.
The gauged su(2)V symmetry is the diagonal combination of su(2)H ⇥su(2) . In ou
cquire vevs (3.2), then the kinetic terms yield the following mass terms for the
= m2
W W+
W +
1
2
m2
AA2
m2
W = g2
f2
+
g2
v2
4
m2
A =
g2
v2
4
.
he massive dark matter W±
= (W1
⌥iW2
)/
p
2 and mediator (dark photon) A
⌧ f2
yields a spectrum where the dark photon is much lighter than the dark m
derivatives with respect to the spin-1 mass eigenstates are
DµH = @µH i
g
p
2
W+
µ T+
+ Wµ T H igAµT3
H
Dµ = @µ i
g
p
2
W+
µ [T+
, ] + Wµ [T , ] igAµ[T3
, ] .
s Mechanism and Leftover Goldstones
linear combination of Goldstone bosons associated with su(2)V . Gauging the
su(2)V promotes this global symmetry to a local symmetry. In unitary gau
cal su(2)V transformation to remove 'V from the theory. It appears solely
polarization of the massive gauge bosons. We may express ' in terms of th
the fields acquire vevs (3.2), then the kinetic terms yield the following mass terms for the g
sons:
Lmass = m2
W W+
W +
1
2
m2
AA2
m2
W = g2
f2
+
g2
v2
4
m2
A =
g2
v2
4
.
e identify the massive dark matter W±
= (W1
⌥iW2
)/
p
2 and mediator (dark photon) A =
e limit v2
⌧ f2
yields a spectrum where the dark photon is much lighter than the dark ma
e covariant derivatives with respect to the spin-1 mass eigenstates are

27
Eaten Goldstones
CCWZ Phys. Rev. 177 (1969) 2247
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generato
H|radial =
1
p
2
✓
0
h
◆
The gauged su(2)V symmetry is the di
the covariant derivatives on the scala
DµH = @µH igWa
µ Ta
s Mechanism and Leftover Goldstones
linear combination of Goldstone bosons associated with su(2)V . Gaugi
u(2)V promotes this global symmetry to a local symmetry. In unita
al su(2)V transformation to remove 'V from the theory. It appears
olarization of the massive gauge bosons. We may express 'V in term
this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA .
aks the u(1) symmetry so that the A eats the only neutral Goldston
e charged states for which there are two pairs of charged Goldstones
d gauge bosons; see Appendix A for an illustrative u(1) example. Fr
rmalized 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
.
ge, '±
V only appears as the longitudinal mode of W±
. The ‘axial’ com
Goldstone boson that remains in the theory. We refer to these as pion
{
combination su(2)V promotes this global symmetry to a local
performs a local su(2)V transformation to remove 'V from the
longitudinal polarization of the massive gauge bosons. We may
by identifying this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W +
Only hHi breaks the u(1) symmetry so that the A eats the on
contrast to the charged states for which there are two pairs of
pair of charged gauge bosons; see Appendix A for an illustrativ
identify the normalized su(2)V Goldstone 'V and the orthogona
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
'±
A =
f
In unitary gauge, '±
V only appears as the longitudinal mode of
is an uneaten Goldstone boson that remains in the theory. We r
them ⇡±
in anticipation of including explicit symmetry breaking
0
EATEN CHARGED GOLDSTONE
ONLY ONE NEUTRAL
GOLDSTONE

28
Eaten Goldstones
CCWZ Phys. Rev. 177 (1969) 2247
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
H|radial =
1
p
2
✓
0
h
◆
DµH = @µH igWa
µ Ta
and Leftover Goldstones
on of Goldstone bosons associated with su(2)V . Gauging the vector
his global symmetry to a local symmetry. In unitary gauge one
rmation to remove 'V from the theory. It appears solely as the
massive gauge bosons. We may express 'V in terms of the 'H,
e kinetic terms:
|2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA . (3.10)
metry so that the A eats the only neutral Goldstone. This is in
for which there are two pairs of charged Goldstones and only one
see Appendix A for an illustrative u(1) example. From (3.10) we
Goldstone 'V and the orthogonal state 'A:
v/2)'±
H
(v/2)2
'±
A =
f'±
H (v/2)'±
p
f2 + (v/2)2
. (3.11)
ears as the longitudinal mode of W±
. The ‘axial’ combination '±
A
that remains in the theory. We refer to these as pions and relabel
{
n su(2)V promotes this global symmetry to a local symmetry. In uni
local su(2)V transformation to remove 'V from the theory. It appea
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 .
reaks the u(1) symmetry so that the A eats the only neutral Goldst
the charged states for which there are two pairs of charged Goldstone
ged gauge bosons; see Appendix A for an illustrative u(1) example.
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, '±
. The ‘axial’ c
en Goldstone boson that remains in the theory. We refer to these as pi
anticipation of including explicit symmetry breaking terms to make th
mmetry Breaking with , 0
EATEN CHARGED GOLDSTONE
u(2)V promotes this global symmetry to a local symmetry. In unitary
al su(2)V transformation to remove 'V from the theory. It appears so
olarization of the massive gauge bosons. We may express 'V in terms
this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA .
ks the u(1) symmetry so that the A eats the only neutral Goldstone.
charged states for which there are two pairs of charged Goldstones an
d gauge bosons; see Appendix A for an illustrative u(1) example. From
rmalized 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
.
ge, '±
. The ‘axial’ comb
Goldstone boson that remains in the theory. We refer to these as pions
ticipation of including explicit symmetry breaking terms to make them
metry Breaking with , 0
ORTHOGONAL GOLDSTONE
d Symmetries
ength g to a two scalar particles: a doublet Hi
and
tion, the su(2) transformation is
(x) ! U (x)U†
, (2.1)
ry matrix and Ta
= 1
2
a
are the generators of su(2)
s respect. a global “flavor” symmetry
= su(2)V ⇥ su(2)A ⇥ u(1)H , (2.2)
Particles and Symmetries
at couples with strength g to a two scalar particles: a d
In this representation, the su(2) transformation is
! UH(x) (x) ! U (x)U†
,
2 ⇥ 2 special unitary matrix and Ta
= 1
2
a
are the gene
ntation.
ctions, the particles respect. a global “flavor” symmetry
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
s transform as
NB: how does this know that 
it should eat the “smaller” vev?

29
Goldstone smorgasborg
EAT
EAT
contrast to the charged states for which ther
pair of charged gauge bosons; see Appendix
identify the normalized su(2)V Goldstone 'V
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
In unitary gauge, '±
V only appears as the lon
is an uneaten Goldstone boson that remains
them ⇡±
in anticipation of including explicit
3.4 Symmetry Breaking with
The simplest form of this model takes only t
V | , 0 =
4!
2 Tr 2
These terms separately break the su(2) and
gauged vector combination of the two and gi
g this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA .
eaks the u(1) symmetry so that the A eats the only neutral Goldston
he charged states for which there are two pairs of charged Goldstones
ed gauge bosons; see Appendix A for an illustrative u(1) example. Fr
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
.
auge, '±
. The ‘axial’ com
n Goldstone boson that remains in the theory. We refer to these as pion
anticipation of including explicit symmetry breaking terms to make them
mmetry Breaking with , 0
form of this model takes only the first two terms in (2.5),
0
UNEATEN
massless modes?  
this sucks.

30
The Model
su(2) : ! U U su(2)H : H ! UHH u(1)H : H ! e H .
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
.
stone modes. The 00
terms and subsequen
ts of the µ and 00
{
{
h ⇠ µf.]
Symmetry Breaking
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
±
⌘
1
+ i 2
p
2
.
hHi =
✓
0
v/
p
2
◆
h i =
1
2
✓
f
f
◆
= fT3
.
Would-be Goldstones
'H ·T p p
h ⇠ µf.]
is
3
p
2 +
2 3
◆
±
⌘
1
+ i 2
p
2
. (3.1)
h i =
1
2
✓
f
f
◆
= fT3
. (3.2)
EAT GOLDSTONES
(MASS TO VECTORS)
EXPLICIT SU(2)A
BREAKING
(MASS TO SCALARS)
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
.
stone modes. The 00
terms and subsequen
ts of the µ and 00
0
term.1
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
H|radial =
1
p
2
✓
0
h
◆
DµH = @µH igWa
µ Ta

31
The Model
gian
e global symmetries of the particle content is
V (2.4)
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
. (2.5)
ental and adjoint of su(2), respectively. We
d H obtain vacuum expectation values (vevs)
eory. This breaking produces a spectrum of
massive gauge bosons. The trilinear µ term
This gives a mass to the remaining the would-
modes of the H and . We systematically
king and 0
terms and subsequently include
terms obeying the global symmetries reduce
b
12⇥2
◆
H =
1
2
|H|2
Tr 2
.
EXPLICIT SU(2)A
BREAKING
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
H|radial =
1
p
2
✓
0
h
◆
DµH = @µH igWa
µ Ta
Invariant under SU(2)V,
but not SU(2)A
Inserting nonlinear fields gives explicit mass term
for the axial Goldstones (pions)
µv
+ µv2/4f 0v2/3
. (3.22)
erm induces mixes the charged Goldstones, '±
and
M2
G =
0
B
@
µv2
4f
µv
2
µv
2
µ
f
1
C
A . (3.23)
0. This corresponds to the massless Goldstone G±
dstone, ⇡±
, has a mass-squared given by the trace:
m2
⇡ = µf
✓
1 +
v2
4f2
◆
. (3.24)
y a rotation

32
Effect of mu
μ encodes post-NLΣM interactiosn 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
<latexit sha1_base64="1vsHceo5oI5Ne6NgSeabgvw7lC4=">AAAB6HicdVDJSgNBEK2JW4xb1KOXxiB4GmZMIPEW9OIxAbNAMoSeTk3S2rPQ3SOEIV/gxYMiXv0kb/6NnQ1U9EHB470qqur5ieBKO86nlVtb39jcym8Xdnb39g+Kh0dtFaeSYYvFIpZdnyoUPMKW5lpgN5FIQ19gx7+/nvmdB5SKx9GtniTohXQU8YAzqo3UDAbFkmOXL8vVSpksSNldkRpxbWeOEizRGBQ/+sOYpSFGmgmqVM91Eu1lVGrOBE4L/VRhQtk9HWHP0IiGqLxsfuiUnBllSIJYmoo0mavfJzIaKjUJfdMZUj1Wv72Z+JfXS3VQ8zIeJanGiC0WBakgOiazr8mQS2RaTAyhTHJzK2FjKinTJpuCCWH1KfmftC9s17HdZqVUv1rGkYcTOIVzcKEKdbiBBrSAAcIjPMOLdWc9Wa/W26I1Zy1njuEHrPcvSkSNQg==</latexit>
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

34
f2
= f2
0 +
3µv2
2 f
v2
= v2
0
µ term causes the vev to shift the H vev, and vice versa.
ing for phenomenological hierarchy
nomenologically we require that the mediator is light and that
matter; this forces
g2
v2
⌧ g2
f2
. µf .
uming g . O(1), we see that the the vev f2
is perturbed by
0 value f2
0 . On the other hand, the hierarchy f, µ v and pe
v2
is shifted by a large amount relative to v2
0. Without loss o
then require that v2
0 is negative and tuned to give a small v2
⌧

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