strongly interacting dark matter at fixed-target experiments€¦ · cmb ⇠ p 3 x f t cmb m dm ⇠...
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Strongly Interacting Dark Matter at Fixed-Target Experiments
ASHER BERLIN
Collaboration with Nikita Blinov, Stefania Gori, Philip Schuster, & Natalia Toro
TeVPA, Ohio State University August 11, 2017
Hidden Sector
Hidden Sector
DM + …
Th
Visible Sector
SM + … + SUSY + …ε ≪ 1
T
100% Hidden
Carlson, Machacek, and Hall (1992)
χ
χ
χ
χ
χ
100% Hidden
Carlson, Machacek, and Hall (1992)
χ
χ
χ
χ
χ KE (Need to expel heat)
100% Hidden
Carlson, Machacek, and Hall (1992)
χ
χ
χ
χ
χ KE (Need to expel heat)
⇒ mχ ≪ keV
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)
m� ⇠ ↵�
�T 2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (2)
m⇡ ⇠ ↵�
�T 2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
pions “⇡” ⌘ ⇡0 , ⌘0 , K0 , K0 , ⇡± , K± (4)
vector mesons “V ” ⌘ ⇢0 , ! , � , K⇤0 , K⇤0 , ⇢± , K⇤± (5)
�(WD ! e+e�) ⇠ ↵em ✏2 ✓2 mWD (6)
(7)
vCMB ⇠p
3xfTCMB
mDM⇠ 10�8 ⇥ 1 GeV
mDM(8)
(9)
mA0 > mDM (10)
(11)
�v / v2 (12)
(13)
Th < m� =) sh ' ⇢�Th
' m�n�
Th/ e�m�/Th (14)
(15)
Sh / e�m�/Th a3 ⇠ constant =) Th ⇠ m�
log a� m�
a(16)
(17)
⇢h ' sh Th ⇠ 1
a3 log a� 1
a3/2 ea(18)
(19)
2Nc
15⇡2 f5⇡
✏µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)
(21)
SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N2f � 1 pions, ⇡a T a (22)
(23)
�(3 ! 2) = n2⇡ h�v2i , h�v2i ⇠
⇣m⇡
f⇡
⌘10 1
m5⇡
(24)
(25)✏
2 cos ✓WA0
µ⌫ Bµ⌫ (26)
(27)
iM ⇠ ✏2 Tr⇥Q2 T a
⇤(28)
(29)
SU(N1)⇥ SU(N2)⇥ U(1)D ⇢ SU(Nf )L+R (30)
(31)m⇡
f⇡⇠ 2⇡ xf
N1/2c
⇥xf/2 + log (
pmpl Teq /m⇡)
⇤ (32)
(33)
eD f⇡mV
A0µ⌫ Tr(QV µ⌫) (34)
(35)
�⇡ > Hf =) sink for entire DM abundance (36)
(37)
�⇡ < Hf =) potential issues with BBN + CMB (38)
(39)
iM ⇠ ✏2 Tr(QT a T b)⇥ Tr(QT b) (40)
Location Timeline Ebeam (GeV) POT Baseline (m)
SeaQuest Fermilab April, 2017 120 1.44⇥ 1018 ! 1020 ? 5� 25SHiP CERN 2026 ? 400 2⇥ 1020 60� 110
Q =(41)
1
90% Hidden
Carlson, Machacek, and Hall (1992)
χ
SM
χ
SM
Th = T
(heat dumped into SM)
χ
χ
χ
χ
χ
{ L � g� h �� }8
<
:
L � g� H ��+ g� HX
f
yf f̄f
9
=
;
8
<
:
L � g� A � i�5 �+ g� AX
f
yf f̄ i�5f
9
=
;
�
L � �c� �2 |H|2
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
M�̃ =
0
B
B
B
B
B
B
B
B
@
0 µ �g2 vu /p2 g1 vu /
p2
µ 0 g2 vd /p2 �g1 vd /
p2
�g2 vu /p2 g2 vd /
p2 M2 0
g1 vu /p2 �g1 vd /
p2 0 M1
1
C
C
C
C
C
C
C
C
A
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
n
H̃u , H̃d , W̃ , B̃o
(1)
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (2)
m� ⇠ ↵�
�
T 2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
(4)
1
The SIMP Miracle
Hochberg, Kuflik, Volansky, Wacker arXiv:1402.5143
π
π
π
π
π
{ L � g� h �� }8
<
:
L � g� H ��+ g� HX
f
yf f̄f
9
=
;
8
<
:
L � g� A � i�5 �+ g� AX
f
yf f̄ i�5f
9
=
;
�
L � �c� �2 |H|2
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
M�̃ =
0
B
B
B
B
B
B
B
B
@
0 µ �g2 vu /p2 g1 vu /
p2
µ 0 g2 vd /p2 �g1 vd /
p2
�g2 vu /p2 g2 vd /
p2 M2 0
g1 vu /p2 �g1 vd /
p2 0 M1
1
C
C
C
C
C
C
C
C
A
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
n
H̃u , H̃d , W̃ , B̃o
(1)
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (2)
m� ⇠ ↵�
�
T 2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
m⇡ ⇠ ↵�
�
T 2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (4)
(5)
1
The SIMP Miracle
Hochberg, Kuflik, Volansky, Wacker arXiv:1402.5143
A Theory of Pions
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)
m� ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (2)
m⇡ ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
pions “⇡” ⌘ ⇡
0, ⌘
0, K
0, K
0, ⇡
±, K
± (4)
vector mesons “V ” ⌘ ⇢
0, ! , � , K
⇤0, K
⇤0, ⇢
±, K
⇤± (5)
�(WD ! e
+e
�) ⇠ ↵em ✏
2✓
2mWD (6)
(7)
vCMB ⇠p
3xfTCMB
mDM⇠ 10�8 ⇥ 1 GeV
mDM(8)
(9)
mA0> mDM (10)
(11)
�v / v
2 (12)
(13)
Th < m� =) sh ' ⇢�
Th' m�n�
Th/ e
�m�/Th (14)
(15)
Sh / e
�m�/Tha
3 ⇠ constant =) Th ⇠ m�
log a(16)
(17)
⇢h ' sh Th ⇠ 1
a
3 log a� 1
a
3/2e
a(18)
(19)
2Nc
15⇡2f
5⇡
✏
µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)
(21)
SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N
2f � 1 pions, ⇡a
T
a (22)
Location Timeline Ebeam (GeV) POT Baseline (m)
SeaQuest Fermilab April, 2017 120 1.44⇥ 1018 ! 1020 ? 5� 25SHiP CERN 2026 ? 400 2⇥ 1020 60� 110
1
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)
m� ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (2)
m⇡ ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
pions “⇡” ⌘ ⇡
0, ⌘
0, K
0, K
0, ⇡
±, K
± (4)
vector mesons “V ” ⌘ ⇢
0, ! , � , K
⇤0, K
⇤0, ⇢
±, K
⇤± (5)
�(WD ! e
+e
�) ⇠ ↵em ✏
2✓
2mWD (6)
(7)
vCMB ⇠p
3xfTCMB
mDM⇠ 10�8 ⇥ 1 GeV
mDM(8)
(9)
mA0> mDM (10)
(11)
�v / v
2 (12)
(13)
Th < m� =) sh ' ⇢�
Th' m�n�
Th/ e
�m�/Th (14)
(15)
Sh / e
�m�/Tha
3 ⇠ constant =) Th ⇠ m�
log a(16)
(17)
⇢h ' sh Th ⇠ 1
a
3 log a� 1
a
3/2e
a(18)
(19)
2Nc
15⇡2f
5⇡
✏
µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)
(21)
SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N
2f � 1 pions, ⇡a
T
a (22)
Location Timeline Ebeam (GeV) POT Baseline (m)
SeaQuest Fermilab April, 2017 120 1.44⇥ 1018 ! 1020 ? 5� 25SHiP CERN 2026 ? 400 2⇥ 1020 60� 110
1
(Wess-Zumino-Witten)
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)
m� ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (2)
m⇡ ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
pions “⇡” ⌘ ⇡
0, ⌘
0, K
0, K
0, ⇡
±, K
± (4)
vector mesons “V ” ⌘ ⇢
0, ! , � , K
⇤0, K
⇤0, ⇢
±, K
⇤± (5)
�(WD ! e
+e
�) ⇠ ↵em ✏
2✓
2mWD (6)
(7)
vCMB ⇠p
3xfTCMB
mDM⇠ 10�8 ⇥ 1 GeV
mDM(8)
(9)
mA0> mDM (10)
(11)
�v / v
2 (12)
(13)
Th < m� =) sh ' ⇢�
Th' m�n�
Th/ e
�m�/Th (14)
(15)
Sh / e
�m�/Tha
3 ⇠ constant =) Th ⇠ m�
log a(16)
(17)
⇢h ' sh Th ⇠ 1
a
3 log a� 1
a
3/2e
a(18)
(19)
2Nc
15⇡2f
5⇡
✏
µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)
(21)
SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N
2f � 1 pions, ⇡a
T
a (22)
(23)
�(3 ! 2) = n
2⇡ h�v2i , h�v2i ⇠
⇣m⇡
f⇡
⌘10 1
m
5⇡
(24)
(25)✏
2 cos ✓WA
0µ⌫ B
µ⌫ (26)
(27)
iM ⇠ ✏
4 Tr⇥Q
2T
a⇤
(28)
(29)
SU(N1)⇥ SU(N2)⇥ U(1)D ⇢ SU(Nf )L+R (30)
Location Timeline Ebeam (GeV) POT Baseline (m)
SeaQuest Fermilab April, 2017 120 1.44⇥ 1018 ! 1020 ? 5� 25SHiP CERN 2026 ? 400 2⇥ 1020 60� 110
Q =(31)
+1
. . .
+1
�1
. . .
�1
0
BBBBBBBBBBB@
1
CCCCCCCCCCCA
N
1
N
2
1
π
SM
π
SM
A’
ε
Th = T
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)
m� ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (2)
m⇡ ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
pions “⇡” ⌘ ⇡
0, ⌘
0, K
0, K
0, ⇡
±, K
± (4)
vector mesons “V ” ⌘ ⇢
0, ! , � , K
⇤0, K
⇤0, ⇢
±, K
⇤± (5)
�(WD ! e
+e
�) ⇠ ↵em ✏
2✓
2mWD (6)
(7)
vCMB ⇠p
3xfTCMB
mDM⇠ 10�8 ⇥ 1 GeV
mDM(8)
(9)
mA0> mDM (10)
(11)
�v / v
2 (12)
(13)
Th < m� =) sh ' ⇢�
Th' m�n�
Th/ e
�m�/Th (14)
(15)
Sh / e
�m�/Tha
3 ⇠ constant =) Th ⇠ m�
log a(16)
(17)
⇢h ' sh Th ⇠ 1
a
3 log a� 1
a
3/2e
a(18)
(19)
2Nc
15⇡2f
5⇡
✏
µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)
(21)
SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N
2f � 1 pions, ⇡a
T
a (22)
(23)
�(3 ! 2) = n
2⇡ h�v2i , h�v2i ⇠
⇣m⇡
f⇡
⌘10 1
m
5⇡
(24)
(25)✏
2 cos ✓WA
0µ⌫ B
µ⌫ (26)
Location Timeline Ebeam (GeV) POT Baseline (m)
SeaQuest Fermilab April, 2017 120 1.44⇥ 1018 ! 1020 ? 5� 25SHiP CERN 2026 ? 400 2⇥ 1020 60� 110
1
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)
m� ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (2)
m⇡ ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
pions “⇡” ⌘ ⇡
0, ⌘
0, K
0, K
0, ⇡
±, K
± (4)
vector mesons “V ” ⌘ ⇢
0, ! , � , K
⇤0, K
⇤0, ⇢
±, K
⇤± (5)
�(WD ! e
+e
�) ⇠ ↵em ✏
2✓
2mWD (6)
(7)
vCMB ⇠p
3xfTCMB
mDM⇠ 10�8 ⇥ 1 GeV
mDM(8)
(9)
mA0> mDM (10)
(11)
�v / v
2 (12)
(13)
Th < m� =) sh ' ⇢�
Th' m�n�
Th/ e
�m�/Th (14)
(15)
Sh / e
�m�/Tha
3 ⇠ constant =) Th ⇠ m�
log a(16)
(17)
⇢h ' sh Th ⇠ 1
a
3 log a� 1
a
3/2e
a(18)
(19)
2Nc
15⇡2f
5⇡
✏
µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)
(21)
SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N
2f � 1 pions, ⇡a
T
a (22)
(23)
�(3 ! 2) = n
2⇡ h�v2i , h�v2i ⇠
⇣m⇡
f⇡
⌘10T
2
m
7⇡
(24)
(25)✏
2 cos ✓WA
0µ⌫ B
µ⌫ (26)
Location Timeline Ebeam (GeV) POT Baseline (m)
SeaQuest Fermilab April, 2017 120 1.44⇥ 1018 ! 1020 ? 5� 25SHiP CERN 2026 ? 400 2⇥ 1020 60� 110
1
(Kinetic mixing)
The SIMP Miracle
Hochberg, Kuflik, Murayama, Volansky, Wacker arXiv: 1411.3727
SU(4), Nf = 3
SU(5), Nf = 3
SU(10), Nf = 3
10-2 10-1 1 100
2
4
6
8
10
10-2
10-1
1
10
102
mπ [GeV]
mπ/f π
SU(Nf )×SU(Nf ) / SU(Nf )
σ scatter/m
π[cm2 /g
]
4
8
12
16
20
The SIMP Miracle
Hochberg, Kuflik, Murayama, Volansky, Wacker arXiv: 1411.3727
mπ/fπ ≫ 1 ⇒ vector mesons nearby in mass mV ~ 4π fπ / Nc
1/2
SU(4), Nf = 3
SU(5), Nf = 3
SU(10), Nf = 3
10-2 10-1 1 100
2
4
6
8
10
10-2
10-1
1
10
102
mπ [GeV]
mπ/f π
SU(Nf )×SU(Nf ) / SU(Nf )
σ scatter/m
π[cm2 /g
]
4
8
12
16
20
A’
Mass Spectrum
Vector Mesons, V
Pions, π
{Prevent π π → A’ A’
{mπ / fπ ≫ 1
~ GeV
~ 100 MeV
Bullet Cluster
(CMB)
A’ DecaysHidden Sector
0 2 4 6 8 10 1210-3
10-2
10-1
1
m pê fp
BRHA¢ Æ
XL
A¢ Decays, m p ê mA¢ = 1ê3mV =1.8 m pmV =1.4 m p
w p0 + V ± p°
r0 h + f h
p + p -
V +V -
π+π- invisible
everything else semi-visible
V Decays
(mV > 2 mπ )
(mV < 2 mπ )
(Model requires mV ~ mπ)
V 0
A′∗
e−
e+V ±
A′∗
e−
e+
π±
V 0, V ±
π0,π±
π0V ±
π±
π0
V 0
A′∗
e−
e+V ±
A′∗
e−
e+
π±
V 0, V ±
π0,π±
π0V ±
π±
π0
V 0
A′∗
e−
e+V ±
A′∗
e−
e+
π±
V 0, V ±
π0,π±
π0V ±
π±
π0
(longer lived)
V Decays
10�2 10�1 100
mV [GeV]
10�5
10�4
10�3
10�2
10�1
100
101
102
103
104
c⌧V
[cm
]
" = 10�3, ↵D = 10�2
V 0
V ±
A′
Vπ
mass
e−
e−
A′
Z
V 0
π0
A′∗
e−
e+
e−
e−
A′
Z
π−
π+
e−
e−
A′
Z
V ±
π∓
A′∗
e−
e+
π±
Signal Examples
missing energy
missing energy + displaced resonant leptons
missing energy + displaced leptons
Parameter Space
Berlin, Blinov, Gori, Schuster, Toro arXiv:171X.XXXX
10-2 10-1 110-810-710-610-510-410-310-210-1
m p @GeVD
e
SIMP, m p : mV : mA¢ = 1 : 1.8 : 3, m p ê fp = p, aD = aem
Wp>WDM
Belle-II
BaBar Hinv.L
BaBar Hvis.L LEP
Hg-2Le,m
LSNDêE137êMiniBooNE
LDMX Hinv.LE137 Hdec.L
ELDER
BulletCluster
Kin. Eq.
Summary
• SIMP cosmology favors mπ / fπ ≫ 1, i.e., mπ ~ mV parametrically true.
• Semi-Visible decays of A’ and V can be tested extensively in low-energy accelerators.
Back Up Slides
Invisible or Visible Decays
10-40 cm
2
10-40 cm
2
10-41 cm
2
10-41 cm
2
10-42 cm
2
10-42 cm
2
10-1 1 101 102 10310-810-710-610-510-410-310-210-11
10-1 1 101 102 10310-810-710-610-510-410-310-210-11
Hochberg, Kuflik, Murayama arXiv: 1512.07917
(upper bound on mA’ from thermalization)
(lower bound on mA’ from CMB)
mA’ mA’
LDMX Reach
10�2 10�1 100 101
mA0 [GeV]
10�7
10�6
10�5
10�4
10�3
10�2
"
HPS 2018
LDMX P2
LDMX inv.
Belle II 20 fb�1
⌦⇡
=⌦
cdm
m⇡ : mV : mA0 = 1 : 1.8 : 3
↵D = 10�2, m⇡/f⇡ = 3
CCD 100g yr
SuperC
DMS
V Decays
(m < 2 mπ )V0
(m > 2 mπ )V±
radiative corrections under
U(1)D
V 0
A′∗
e−
e+V ±
A′∗
e−
e+
π±
V 0, V ±
π0,π±
π0V ±
π±
π0
V 0
A′∗
e−
e+V ±
A′∗
e−
e+
π±
V 0, V ±
π0,π±
π0V ±
π±
π0
LDMX Reach
10�2 10�1 100
mA0 [GeV]
10�7
10�6
10�5
10�4
10�3
10�2
"
E137
OrsayBaBar inv.
LSND
LDMX inv.
10cm
2/g
1cm
2/g
⌦⇡
=⌦
cdm
HPS 2016
HPS 2018
LDMX P1
LDMX P1
↵D = 10�2, m⇡/f⇡ = 3
π
π
π
V
Forbidden Semi-Annihilation
m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)
m� ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (2)
m⇡ ⇠ ↵�
�T
2eqmpl
�1/3 ⇠ ↵� ⇥ 1 GeV (3)
pions “⇡” ⌘ ⇡
0, ⌘
0, K
0, K
0, ⇡
±, K
± (4)
vector mesons “V ” ⌘ ⇢
0, ! , � , K
⇤0, K
⇤0, ⇢
±, K
⇤± (5)
�(WD ! e
+e
�) ⇠ ↵em ✏
2✓
2mWD (6)
(7)
vCMB ⇠p
3xfTCMB
mDM⇠ 10�8 ⇥ 1 GeV
mDM(8)
(9)
mA0> mDM (10)
(11)
�v / v
2 (12)
(13)
Th < m� =) sh ' ⇢�
Th' m�n�
Th/ e
�m�/Th (14)
(15)
Sh / e
�m�/Tha
3 ⇠ constant =) Th ⇠ m�
log a(16)
(17)
⇢h ' sh Th ⇠ 1
a
3 log a� 1
a
3/2e
a(18)
(19)
2Nc
15⇡2f
5⇡
✏
µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)
(21)
SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N
2f � 1 pions, ⇡a
T
a (22)
(23)
�(3 ! 2) = n
2⇡ h�v2i , h�v2i ⇠
⇣m⇡
f⇡
⌘10 1
m
5⇡
(24)
(25)✏
2 cos ✓WA
0µ⌫ B
µ⌫ (26)
(27)
iM ⇠ ✏
4 Tr⇥Q
2T
a⇤
(28)
(29)
SU(N1)⇥ SU(N2)⇥ U(1)D ⇢ SU(Nf )L+R (30)
(31)m⇡
f⇡⇠ 2⇡ xf
N
1/2c
⇥xf/2 + log (
pmpl Teq /m⇡)
⇤ (32)
Location Timeline Ebeam (GeV) POT Baseline (m)
SeaQuest Fermilab April, 2017 120 1.44⇥ 1018 ! 1020 ? 5� 25SHiP CERN 2026 ? 400 2⇥ 1020 60� 110
Q =(33)
+1
. . .
+1
�1
. . .
�1
0
BBBBBBBBBBB@
1
CCCCCCCCCCCA
N
1
N
2
1
+1
. . .
+1
�1
. . .
�1
0
BBBBBBBBBBB@
1
CCCCCCCCCCCA
N1
N2
⇡aT a ⇠
0
@N1 ⇥N1 N2 ⇥N1
N1 ⇥N2 N2 ⇥N2
1
A ⇠
0
@(1, 1, 0) + (A, 1, 0)�N1, N2,+2
�
�N1, N2,�2
�(1, 1, 0) + (1, A, 0)
1
A
h�vi ⇠ e�(mV �m⇡)/T
m2⇡
⇠ e�(f⇡�m⇡)/T
m2⇡
� ⇠ n⇡e�(f⇡�m⇡)/m⇡
m2⇡
⇠ H ⇠ m2⇡
mpl
n⇡ ⇠ m4⇡
mple(m⇡/f⇡)
�1�1
⇢eq ⇠ m4⇡
mple(m⇡/f⇡)
�1�1
✓Teq
m⇡
◆3
⇠ T 4eq
m⇡
f⇡⇠
✓1 + log
(Teq mpl)1/2
m⇡
◆�1
(42)
2
Forbidden Semi-Annihilation
10-2 10-1 11
5
10
4p
m p @GeVD
mpêf p
Semi-Annihilation
mVêm p >>1
mVêm p =
2.2
mVêm p =
2
mVêmp = 1
.8
mV êmp = 4 p
Hmp ê fpL Nc1ê2
relaxes self-interaction constraints
3→2
DecaysHidden Sector
Here, for reference, we list the relevant decays in the specific model under consideration, using the KSFRrelation when applicable:
�(A0 ! `+`�) =↵em ✏2
3
�1� 4 r2
`
�1/2 �1 + 2 r2
`
�m
A
0
�(A0 ! hadrons) = R(ps = m
A
0) �(A0 ! µ+µ�)
�(A0 ! ⇡⇡) =2↵
D
3
(1� 4r2⇡
)3/2
(1� r�2V
)2m
A
0
�(A0 ! ⌘0 ⇢) =↵D
r2V
256⇡4
✓m
⇡
/f⇡
r⇡
◆4 h1� 2(r2
⇡
+ r2V
) + (r2⇡
� r2V
)2i3/2
mA
0
�(A0 ! ⌘0 �) =↵D
r2V
128⇡4
✓m
⇡
/f⇡
r⇡
◆4 h1� 2(r2
⇡
+ r2V
) + (r2⇡
� r2V
)2i3/2
mA
0
�(A0 ! ⇡0 !) =3↵
D
r2V
256⇡4
✓m
⇡
/f⇡
r⇡
◆4 h1� 2(r2
⇡
+ r2V
) + (r2⇡
� r2V
)2i3/2
mA
0
�(A0 ! K0 K⇤0 , K0 K⇤0) =3↵
D
r2V
128⇡4
✓m
⇡
/f⇡
r⇡
◆4 h1� 2(r2
⇡
+ r2V
) + (r2⇡
� r2V
)2i3/2
mA
0
�(A0 ! ⇡± ⇢⌥) =3↵
D
r2V
128⇡4
✓m
⇡
/f⇡
r⇡
◆4 h1� 2(r2
⇡
+ r2V
) + (r2⇡
� r2V
)2i3/2
mA
0
�(A0 ! K± K⇤⌥) =3↵
D
r2V
128⇡4
✓m
⇡
/f⇡
r⇡
◆4 h1� 2(r2
⇡
+ r2V
) + (r2⇡
� r2V
)2i3/2
mA
0
�(A0 ! V V ) =↵D
6
(1� 4r2V
)1/2(1 + 16r2V
� 68r4V
� 48r6V
)
(1� r2V
)2m
A
0
�(⇢ ! `+`�) =32⇡ ↵em ↵
D
✏2
3
✓r⇡
m⇡
/f⇡
◆2
(r2V
� 4r2`
)1/2 (r2V
+ 2r2`
) (1� r2V
)�2 mA
0
�(� ! `+`�) =16⇡ ↵em ↵
D
✏2
3
✓r⇡
m⇡
/f⇡
◆2
(r2V
� 4r2`
)1/2 (r2V
+ 2r2`
) (1� r2V
)�2 mA
0
�(! ! `+`�) = 0
�(V ! hadrons) = R(ps = m
V
) �(V ! µ+µ�)
�(! ! `+`�⇡0) ⇠ �(⇢± ! `+`�⇡±) ⇠ �(K⇤± ! `+`�K±) ⇠ ✏2↵D
m5V
/m4A
0 . (131)
In the last line of the box above, we have given a parametric form for the 3-body decay of vector mesons,which is especially relevant for the vectors that are unable to mix with the photon. In the limit that m
`
' 0,
22
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